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in the sense that two telegraph signals (in other words, two electric currents) can travel in opposite directions on the same wire, at the same time, without interfering with each other

This is the point. A telegraph wire is not just a wire that transports a direct current. It transports a signal, which in essence is a voltage or current wave. The current oscillates around a value of $0~V$ (or $0~A$).

If two current waves travel in the opposite direction, the waves have no trouble passing through each other, just as two sound waves are able to travel in opposite directions in the same medium.

(Here, the blue travels to the left, green to the right, and the red wave is the resulting superposition of them. The red wave is the current/voltage distribution that is measured in the wire over time.)

Instead, the value that you would measure at any point corresponds to the linear sum of the amplitudes of both waves - that is what is called the Principle of Superposition. Waves obey this principle if their underlying equations are linear, i.e. the wave equation doesn't contain higher powers ($x^2$, $x^3$, etc) of the variable that is "waving". That would be the voltage $U(x, t)$ or the current $I(x,t)$ in the telegraph case.

Specifically, the equations describing waves in a transmission line even have a name: the Telegrapher's Equation.
$$frac{partial}{partial x} U(x,t) = -L frac{partial}{partial t} I(x,t) - R I(x,t)$$
$$frac{partial}{partial x} I(x,t) = -C frac{partial}{partial t} U(x,t) - G U(x,t)$$

$L, C, G $ and $R$ are properties of the transmission line (inductance, capacitance, conductance and resistance) and determine how the shape of the signal is distorted, attenuated and slowed down while it is traveling down the line. The rest describes how an excitation of the voltage and current propagate through this transmission line along the position $x$ and through time $t$.

As you can see, all of the quantities of interest (voltage $U$ and current $I$) appear only in the first power. That has the consequence that if you find two solutions to the wave equation, then also their sum will be a solution to the same equation (even when multiplying them by a factor). So if $U_1(x,t)$ and $U_2(x,t)$ are solutions to the wave equation, then also $$U(x,t) = alpha cdot U_1(x,t) + beta cdot U_2(x,t)$$ is a solution, where $alpha$ and $beta$ are factors.

Side note about DC:

Having two currents flow in opposite directions would cancel out their contributions and result in no current. Alternatively, you can convince yourself that a direct current (DC) cannot flow in both directions at once, simply by Ohm's Law:

Say you have a wire with a certain resistance $R$. According to Ohm's Law, we need a potential difference of $U = varphi_2 - varphi_1$ between the ends to cause a current $I = frac{U}{R}$ to flow. The current (composed to moving positive charges) moves from the more positive potential to the more negative potential.

We can now flip the potentials: $$U' = varphi_1 - varphi_2 = -U.$$ But since the resistance stays the same, this also flips the current: $$I' = frac{U'}{R} = frac{-U}{R} = -I.$$

If we make both potentials equal, there will be no difference and the current will be zero.

The only way to have a current come out at both ends is to have a source in the middle, which is not really interesting.

There's your problem: telegraph signals are not electric currents. (We might as well say that telegraph signals are voltage instead.) Which is correct? Neither.

To solve this, give up on electronics and instead fall back to the physics behind it. Actually the telegraph signals (and even all electrical signals everywhere) are actually electrical energy; same thing as light and radio waves. Signals are changes, and a changing current involves voltage, same as changing voltages involving current. Signals are the watts, not just amperes and not just volts.

Signal-energy behaves differently than currents in circuits. While the energy zips across a circuit, instead the amps or charge-flow does not. Charges just rotate through the loop as a whole, or perhaps wiggle slightly back and forth, but current doesn't fly forwards at lightspeed. Something does fly along at lightspeed though. We measure it and discuss it in terms of watts or "wattage." The amps don't fly fast, the amps are different, amps are the slow motions of the "medium;" that charge-sea found inside every wire. Waves versus medium. Somewhat like sound waves versus wind. Electric current is like wind, while signals are like sound waves. (And of course sound waves are back-and-forth wind! The air wiggles, while the waves propagate forwards.)

How can two independent signals pass across an electric circuit? First ask yourself how two independent sound waves are able to pass through the same region of air. And on a pond, throw two pebbles and ask yourself how two bullseye ripple-patterns pass through each other without interacting. Why doesn't one laser beam block another one whenever they cross? It's just something that all waves can do, if the medium is linear. In a linear system, waves can add and then subtract again, so they cross each other without interacting. It works for light inside an optical fiber. It works for sound inside an organ-pipe. It works for coax cable with pulses going in opposite directions, and it works for telegraph signals propagating at lightspeed across a single pair, a single circuit.

The answer to your question involves the waves-chapter of your physics book. The answer to your particular circuits question opens up an entire fascinating field of electronics: cable reflections and standing waves on wires.

On the other hand, two direct currents cannot occupy the same circuit, since they lose their identity, combining to form a sum-current. (Don't forget that every circuit is a one-turn inductor. Similarly, two different voltages cannot occupy the same capacitor! In both cases they combine, and cannot be un-subtracted again.) Two direct currents can occupy a single wire, whenever that wire is a common section of two otherwise separate circuits. But they do this by adding up to form a third current within that common section. (For example, they might subtract to zero current in that section, if they happened to be equal and opposite. One electron can't actually flow in two directions simultaneously.)

Yet at the same time, two completely independent energy-waves (signals) can propagate across one single circuit. HOW? It involves E and M both, and that contains the secret: to understand it we must look at both wires of the long pair, and we must include voltage as well as current. Your question cannot be answered as long as we concentrate on single wires and currents alone, while ignoring the two wires and the voltage across them.

In a single circuit, current is a closed circle, like a flywheel. It doesn't start out in one place and flow to another (instead it just goes clockwise, CW, or perhaps CCW, much like a drive-belt.) A current in a circuit is like a spun flywheel, a closed loop. But something sure does go one-way, right? Whenever a battery lights a bulb, something must go from the battery to the bulb and not return back to the battery. That something is not current. Instead it's EM energy, where energy-flow is measured in terms of watts; of volts times amps. In a flashlight circuit, the wattage is a fast one-way flow from battery to bulb. But the current is a very slow circular flow. Again, the "signal" going from battery to bulb is made of EM energy, not amperes and not electrons.

So here's the start of your answer: in a single circuit, how can we know which direction the electrical energy is flowing? Simple: look at the value of wattage. Specifically: multiply the volts between the wires times the amps through them. If the result is positive, then energy is flowing in one direction, and if it's negative then it's flowing in the other. With a flashlight, hook up your voltmeter and ammeter so they give positive wattage when we multiply them together. Then, when you remove the bulb and install a battery charger instead, the current reverses, so we have energy flowing backwards, into the battery. (This idea is critical with AC, where if V and I waves are in synch, the energy flows continuously forward, but if V and I are at 180deg, the energy instead flows backward.)

So, on a long cable, with an electrical pulse which has positive wattage, the pulse is zooming along left, while if wattage was negative, the pulse is going right. If we suddenly connect and disconnect the flashlight battery, we're launching an energy-wave off along the two wires. It travels at lightspeed and gets absorbed by the flashlight bulb, which lights up. If we leave the battery continuously attached, then still an energy-wave flows to the bulb, even if there's no ripples at all. That's the first concept in basic waves-engineering: the propagation of electrical energy across circuits ...and the idea that "DC" is really just "AC" at very low frequency.

Back to the start again: how can two signal-pulses fly in opposite directions along the same wire-pair? (Note that it must be a wire pair with volts included. Not a single wire.) It can occur if one of the pulses has positive wattage and goes left, while the other pulse has negative wattage and is going right. One pulse might be composed of positive volts and positive amps, while the other pulse is made from negative volts and positive amps. Both pulses are EM waves.

PS

Aha, I see another approach! (Ignore it if you wish, since sheesh this is long.) Suppose we have two separate circuits, two flashlights, but then we meld together one short wire-section from each? The two circuits have one piece of wire in common. Do they interact? Nope, because inside the common wire, the currents just add and subtract again. Each battery lights up its own bulb independently, because each circuit-loop has its own separate battery-voltage and its own separate loop-current. Yet in that common wire, it seems as if two different electric currents are flowing! They're not, not really, because one "circuit current" is the current in one entire loop, including one battery, bulb, and entire closed ring of conductors. In that combined wire, the two currents get added at one end of the wire, then subtracted again at the other. The two energy-waves in each circuit remain independent, even though the currents in their common wire can add and subtract.

This shows us that the answer to your original question cannot involve a single wire. It only can be answered by backing off and taking a wider view; by also including the voltage across two wires.

This also shows how "linear" versus "nonlinear" works. In the common wire, at one end the two currents have combined by adding together. But then they perfectly subtract apart again at the other end. This lets the two loops remain independent. But what if this didn't happen, and instead the currents in the single wire weren't a simple sum combination? Aha, that would be "NONLINEAR." In that case we couldn't cleanly separate them once combined. The "adding together" at one end of the wire wouldn't be perfectly equal to the "subtracting apart" at the other end, and in that case the two separate circuits would start interacting. One battery would begin lighting the other bulb slightly. The signals of the two circuits would truly mix together.

PPPPS

This sort of question has a long history, and a popular book about it is THE MAXWELLIANS, by BJ Hunt. The infamous Oliver Heaviside figured out that telegraph signals were actually EM waves, but then he was nearly suppressed by William Preece, head of the UK govt telegraphy office, who "knew" that dots and dashes were simply currents, period, end of story and don't ask questions or WH Preece will make you sorry! :) Heaviside used his new EM theory of cable-waves to solve a huge telegraphy problem: for any signals traveling along 100KM telegraph lines, the dots would either disappear or "ripple," and for telephone lines, long distance transmission was completely distorted and impossible. (The problem was found to be wave-dispersion or "chirp," where the low freqs travel faster than the high.) Heaviside's "telegrapher's equation" and his "loading coils" fixed this, letting telegraphy become broadband, even when over immense distances. He single-handedly created long-distance telephone. But Preece quickly halted this heresy by using his political power to start an anti-Heaviside bad-mouthing campaign in the press, and a whispering campaign among engineers. Then in the USA, Pupin of Columbia pretended to invent Heaviside's loading coils, patented them, and made millions via Bell Telephone, while Heaviside remained nearly penniless, not gaining fame until after he'd died. (Heh, a Tesla/Marconi story long before Tesla and Marconi. Pupin even played a big role in Tesla's downfall!) So now you see why I'm in love with the story of telegraph-is-EM-waves. Obsessed. Don't even get me started! Oops, too late. :)

Andreas H mentioned the circulator for waveguides. In analog phones this job is done by an imperfect hybrid circuit called the Anti sidetone induction coil (ASTIC). A perfect hybrid coil would transmit and receive speech concurrently and separately, ie the signal from your transmitter would travel through the wires to the receiver at the other end and the signal from the distant transmitter would travel to your receiver on the same pair of wires. It was realised early on that people needed to hear themselves speak so the ASTIC allows part of the signal from the local transmitter to pass across to the local receiver.

Within a local analog exchange area the circuit would be two wires all the way from one phone, via the relays in the exchange to the other phone. Once you start travelling between exchanges the signal would be split by a hybrid coil at the exchange and speech in one direction would travel on a different circuit to speech in the other direction (4-wire junction circuit). This allowed the speech to be amplified since amplifiers are unidirectional (one way only). At the remote exchange the two separate paths would be recombined by a hybrid coil and the last leg of the call would be on a pair of wires.

Speech on analog phones and exchanges was 300Hz to 3400Hz so this is low frequency EM waves.

However if you are transferring power, either AC or DC, then we don't have different currents going different ways in the same wire. For example, in a particular state the energy supply companies are required to supply a percentage of 'green' energy but they do not have sufficient 'green' generation resources so they buy the energy from out of state. At the same time they sell surplus non-green energy out of state. If they are buying and selling energy over the same interconnect (wires) then there isn't two competing power flows travelling in opposite directions on the same wire. If State A is buying 500MW of capacity from State B and State B is buying 400MW of capacity from State A then there is be a 100MW flow from State B to State A. The accounting might say 500MW and 400MW but the electrical reality is 100MW. State A will pay for 500MW of green energy but it will be 400MW local non-green plus 100MW out-of-state green.

They do interfere.

Electric signals travel down wires like waves on water. and when two waves meet you get inteferance.

But because the wires are linear, the interferance takes the form of addition, and so it is not destructive oth the info0rmation, and thus if you know what one of the signals is you can find the other signal by subtraction.

Telephone lines use (used?) a circuit called a hybrid that isolates the incoming and outgoing signals allowing a single copper circuit to carry voice signals in both directions.

The telegraph probably used something similar having the sender subtract its own signal from what it sees on the line, allowing it to determine what was arriving from the other end simultaneously with transmitting its own signal.

It is not current, but SIGNAL that travels in any or all directions. This is why a telephone handset doesn't have to interrupt the received sound when you speak, and
that is more familiar to us than telegraph protocols.

This is a bit of gimmickry, called a 'hybrid', that presents a signal to your
ear that has mainly the signal from the distant phone, and
creates signal (modulation of current) according to your voice applied to
the microphone. What you HEAR isn't the 'current in the wire' which
is equally modulated by two voices, it's 90% the distant voice that you
hear, and only 10% of your own. A similar hybrid at the other
end of the connection cancels the main part of his/her vocal input
so that your voice is heard strongly in that phone receiver.

The hybrid is a signal-adding circuit, that has access to both your voice
and the combination of two voices (on the line), and combines
them to reinforce the message from a distance. Nothing in this
scheme is unavailable to a telegraph office, which can also operate as a
receive station even while it is transmitting.

It is NOT easily available
to a wireless transmitter (the non-digital kind) which would usually
have a push-to-talk interrupting switch. Our cellphones sending
digital packets are doing a lot of interrupting, just fast enough
that it rarely bothers us, because that hybrid function interacts
badly with a receiver that overloads while a transmission is in progress.

You wrote:

How is it that two electric currents can travel in opposite directions on the same wire, at the same time, without interfering with each other?

but the original text says:

two telegraph signals can travel in opposite directions on the same wire at the same time without interfering with one another

Here's the contradiction: a telegraph signal and an electric current are not the same thing. The electric current is the linear superposition of waves set in motion on the line by the transducers at each end. The current at one instant at one point on the line can have only one value, but we can calculate that value by calculating the contribution of the waves from the signals imposed on each end of the line and adding them together.

As a simpler but directly observable system, consider a stereo playing music in a room. One speaker does not change how the pressure waves from the other speaker propagate. The net pressure gradient at any point in space and instant is the the result of adding of the pressure waves from each speaker.

Even though physical quantities like current or pressure can have only one value, if we know that those quantities are influenced by an additive combination of causes, the linear superposition principle allows the system to be broken into smaller parts which can be considered separately: in this case the telegraph station at each end of the line, and the waves it generates which propagate down the line.

Signals are made up of waves. Waves pass each other, and after the passing are unchanged. Electromagnetic waves. Waves on the sea also pass each other (although they sometimes do have effects which I won't get into). "Interfere" was a bad choice of word by the author. Nobody can really tell you why. But you already instinctively know that waves can pass each other. Just think about light shining out of a window and in through a window at the same time. This does not seem to be perplexing, does it?

In your question you use the word "current." Currents are another matter. Current in a wire is basically defined as the flow of charge past a point. This would be net flow. So it does not make sense to talk about currents somehow passing each other.

I am trying to avoid talking about more advanced transmission line effects such as capacitance and inductance because I am afraid it will just muddy the waters even more. The bottom line is that signals can pass each other, and during the passing, at the location of the passing, they effect each other. But after the passing, they continue on as if the passing never happened. Just think of light passing both ways through a window.

Your analogy is broken. Don't think of a lane of cars, unless you think of the cars as bumper cars all put together.

The actual overall average speed of electrons moving through a wire is pretty slow. The drift velocity of electrons in a wire is typically several micrometers/second, not fast at all.

What propagates through the wire travels from electron to electron, from source to destination. That process happens very quickly, nearly the speed of light. In the highway analogy, it would be analogous to hitting the first car, and each car bumping the one in front of it. Despite each car moving slowly overall, a wave could propagate through the chain provided you can hit it hard enough.

Multiple sound waves can obviously travel through the air in multiple directions simultaneously. However, when you shout something out, a single molecule doesn't necessarily travel directly from your mouth to the listeners ear. Instead, the bouncing between molecules through the air is what transmits the sound. The same thing is basically true for electrical signals.

Consider the following situation:

Lets assume we have a single wire pair with a controllable voltage source at one end and a controllable current sink at the other. Since both ends can measure the signal of the other end (at the voltage source we can measure current and at the current source we can measure voltage) we can transmit information in both directions. There is no frequency or time multiplexing involved. And there is no interference and we do not need to invoke wave theory.

More details are in my answer on Physics SE.

An aerial cable for a satellite dish carries currents in two directions - An 18 Volt DC signal is supplied by the tuner to power the LNB at the focal point of the dish, and at the same time the LNB sends a signal of 4-12 GHz back down to the tuner, over the same wire.

Both are electric currents, but one is DC and is flat, the other is Radio Frequency and varies.

Its because any wave can pass through each other. Interference will occur, but this does not stop waves.

Its like asking why two waves in a pond can pass each other. If the waves are completely opposite, they will annihilate, otherwise they will weaken each other and continue on.

It has been observed, by many engineers and reasearchers (including myself) that metal conductors have linear behaviour with regards to electric currents and voltages. However, as with most materials, linear behaviour only exists over a certain range. High levels of current will result in non linear behaviour. With good conductors, like copper, silver and gold, the range of linear behaviour is quite large. These metals have low (but not zero) resistance. (If you assume metals have zero resistance, then you will end up with strange predictions that do not match reality)

At low current density, there are lots of free spaces in the metal for electons to move into, and they don't bump into each other or get stuck very often. so not much energy is absorbed by the metal and behaviour appears linear ( the bumper cars are far appart)

When the current density in the metal becomes high enough, then the current transfers significant energey to the metal, which changes it s resistance, and the behvaiour has become non linear. As simple example is hooking a thin wire (like 28 gauge) across the terminals of a large 12V car battery. The metal becomes hot, eventually it melts and breaks the circuit. This is VERY nonlinear behaviour. That wire is probably carrying 50 Amps or so. (DO NOT try this yourself - you can get bits of molten metal flying around, can cause fires and severe damage to your eyes) On the other hand if I put two signals on this same wire (before I melt it) with 0.001 amps each, behaviour will be quite linear.

This guy is armwaving to make a more existential point. It works in principle, but not like he says. And with signal, not current.

Heck, even in radio, two transmitters can and do block concurent usage. Listen to this at 1:25. That "Booooop" is both airplanes acknowledging "their" takeoff clearance, but stepping on each other so at least one is not heard.

If you are using a DC telegraph system, same problem. If either telegraph key pressed down, it will cause both sounders to activate. It really isn't possible to send DC signals in opposite directions in the DC domain, (except via a CSMA-CD style courtesy of waiting until the other person is finished, and being wary of two people starting up at once).

However, imagine if telegraph station 1 transmits DC, and telegrph station 2 has its sounder connected through an AC-blocking choke. Station 2 transmits by switching 1000Hz AC on and off, which only station 1 can hear because its sounder has a right-sized capacitor, which passes 1000 Hz AC but blocks DC.

You can expand this to multiple AC frequencies by using "bandpass" filters which only allow through a certain frequency. Consider the bah-boo-BEEP tone that hails the fire station on the TV series Chicago Fire. That show is a huge tribute and shout-out to a 1970s show called Emergency, the origin of the tones. Emergency portrays a 1960s-era fire dispatching system where multiple frequencies were used in just that fashion.

Two stations transmitting at once simply create a chord on the wire. The frequencies have to be chosen smartly so the chords do not interfere with each other.

All the stations hear all the signals. They just disregard "their own dogfood", i.e. the signal they are transmitting.

It can get more complex, with carrier waves being modulated. At this point we are talking about radio spectrum, but on a wire.