Professional astronomers don't really look through telescopes any more. Even leaving aside all the observations that look at frequencies invisible to the human eye, with the advent of photographic plates it became possible to see much fainter stars with the same telescope by taking a minutes- or hours-long exposure than by actually looking through it. (To the right you see the Leviathan of Parsonstown, the last giant telescope built in the pre-photographic era.) CCDs, more sensitive and more consistent, have now replaced photographic plates. So most observations are just long exposures that go onto disk for later processing. That raises the question: do you actually need to be at the telescope during your observation?
In many cases, at least in radio astronomy, we don't go to the telescope at all. For some telescopes, like the VLA, when you are allocated time, you simply send them an observing script - "look at my target for five minutes, then the calibration source for two, then back to my source, then..." - and the operators run it and send you an email when your data's available. For others, for example the GBT and Arecibo, you have to babysit the observation remotely.
For this sort of telescope, about half an hour before the observation starts, you open up a VNC session on one of the computers at the telescope, and you give the operator a call. (At Arecibo the automated attendant gives you a cheery message "Este es el observatorio de Arecibo!")
The operators at these telescopes are generally non-astronomers, locals put in charge of the telescope. They are responsible for safety, both of people and of the telescope. At the GBT, on the control room balcony they have a pair of high-powered Navy surplus binoculars so the operator can take a good look at the telescope, make sure nobody's nearby and there's nothing wrong with it. Their biggest concern is weather: imagine the effect of a sturdy gust of wind on a 100-meter dish. So it's their job to handle "wind stows", tipping the telescope up until the dish is horizontal ("birdbath position") to reduce the wind load, and other weather concerns (one observation I had scheduled, the operator had to take the telescope out of wind stow every so often to dump the snow out of it - as you might imagine, I didn't get to actually observe that day).
The other job of operators is to control access to the telescope. Normally telescope time is allocated by the telescope access committee, but it's the operator who actually hands over control of the telescope pointing to the astronomers who have time. Normally this is perfectly routine, though sometimes someone will finish early (perhaps because their source set) or ask for a few extra minutes (perhaps to re-do a calibration scan). But it also happens that the schedule gets changed at the last minute. For example, at Arecibo they have a megawatt radar transmitter they use for atmospheric and planetary studies. This runs on its own generator, and when something breaks, they can't use the time and suddenly a slot opens up in the schedule. Usually the operator calls somebody they know wants time, or offers the time to whoever has the time block before or after. In any case, it's the operator who hands you control of the telescope when your time starts.
These big telescopes are very complicated machines, and so they each have a collection of software used to control them. This handles pointing and tracking, receiver and signal path selection, gain adjustment and balancing, and in most cases, control of the recording instrument. So normally, you command the telescope to point to your source, you set up and balance the receivers (usually using a standard script) and you hit "start observation" and the telescope starts taking data.
This system is necessarily all sui generis hardware and software, cobbled together and more-or-less tested. It normally works exceedingly well, but problems do crop up. So it's good, when possible, to come up with some way of monitoring the data that's coming in. If you're looking at a known pulsar, the usual answer is to fold a short piece of the data at the pulsar period and see if you can see the profile. If that comes out okay, things are probably fine.
For pulsar astronomy, though, hardware has been improving rapidly. We now have new backends - recording instruments - that can do much better timing than was possible even a few years ago. These systems are complicated experimental software running on clusters of computers, and they have their quirks; among others, they're not integrated with the standard observing software. So for these systems, you have to open up another desktop, run a handful of arcane unix commands, and hope nothing goes wrong you don't know how to fix (see above screenshot for what it looks like when things are working fine). But the data that comes out is beautiful: we can measure pulse shape and arrival time with microsecond accuracy.
So remote observing basically amounts to logging in to a VNC session on a computer at the telescope, using telescope control software to get the system pointed and calibrated, and then starting up the observing system. Is this losing something compared to actually going to the telescope and observing? Yes and no. Control-wise, if you're sitting in the control room, you log into exactly the same VNC session and use exactly the same software. And if you're observing for an hour and a half every three weeks, it saves an awful lot of travelling for anyone who doesn't live at the telescope. But if you're at the telescope you can talk to the operators face-to-face, and there are astronomers and telescope experts on-site you can discuss things with in person. It's also probably easier for the operator to reach you if something comes up (like a free block of time).