For reference I'm watching the tv show Deadwood. In season 2 a telegraph operator shows up but he's only one person. Would there be certain times that a telegraph would be expected? How would he be able to always get the telegraphs? I feel like at a later point a town might have multiple people monitoring the telegraph, but how does it work before that when the telegraph in that area is first set up?
Day telegrams were a standard rate; night telegrams were cheaper because the line was less busy, but there was also the understanding that the person at the other end might not be awake. Not being a live telephone call, it was possible to handle any messages that arrived in the morning.
Working hours depended on the office; in a larger Western Union office there were multiple shifts, with a regular shift being 10 hours and a night shift being seven and a half. Women were not expected to work nights (this was in contrast with "hello girls" manning telephone switchboards once those started) although it could happen in some circumstances (like when making up time lost due to illness).
Out where there were solo operators -- your Deadwood situation -- they were generally regular business hours. They might have days with no work or days where they work constantly (in which case they could go past your typical 10-hour day in order to handle all the messages).
I believe the TV show was set before Deadwood had a train, but telegraph operators for trains could have their hours vary quite a bit, as an operator might have to stay up late if a train was late.
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Jepsen, T. C. (2000). My Sisters Telegraphic: Women in the Telegraph Office, 1846-1950. Ohio University Press.
Telegraph operation was not treated as as mechanized communication channel like, say, email, which we expect to be transmitted and received any time. It was, given that the transmitting and sending devices were people, still a very human communication channel. In spite of the medium being rather abstract and mechanical, telegraph communication was still a conversation between two people. Some operators would get a reputation for being chatty, called "rag-chewing", and develop friends in other places that they'd never seen nor even heard.
So, rather like if messages were being passed on the telephone, telegraph operators would check if there was someone on the other end of the line, and if not would store the message for later. They would also probably know the habits of other stations with whom they could connect; big ones would be open around the clock, small ones might have normal-ish hours. Railroad telegraph stations would plan to be there around the schedule of incoming trains.