I was actually going to ask a similar question not too long ago. How did messages get routed to a specific location? Were all telegraph operators listening to the same chatter across the wires? Basically how did the telegraph “network” work?
Every operator on the same line could listen in to all the traffic going on on it, which required rules (either formally formalised by the telegraph company, or just unwritten understandings by the operators, who had a strong esprit de corps and all knew each other) that you wouldn’t cut in on someone else’s traffic. Most traffic was business related and hence city-to-city, so city operators would take priority over rural ones. Some stations (again, mostly in cities) served as hubs where several branch lines met, and in such offices, a message coming in from one line but destined for the other would be transcribed by an operator on the incoming line, physically carried over by a messenger to an operator working the other line, and re-sent.
That’s at least how it worked in the beginning. Remember that telegraphy was a major thing for over a century, so technology was improved over time. One such improvement was the multiplexer that has been mentioned already. Another were paper tape controlled automatic telegraphs whereby operators would not work the line directly but punch messages onto a tape of paper, which would be fed into an automatic reader that could key messages faster than a human could.