The tail light at the rear is usually already set to auto in timetable mode. You can't set it from the front cab as the switch there only controls the front lights so you've got to go to the rear cab if they're not already set.
Then what does the taillight switch in the front cab do? I usually just set it to auto assuming that'd turn on the rear lights on the rear, but I'm probably wrong :P
As matjamca said that's for setting the tail lights at that end. Setting it to auto means that when you key out of that cab the tail lights turn on automatically.
Does automatically mean that they only come on when the front cab has its master key inserted, or perhaps when it’s headlights are in operation?
In my experience the master key is only needed to actually get the train moving (so throttle/brake controls). As far as I've seen, headlight controls, as well as other light controls (cab lights, passenger lights, instrument lights, destination board lights) can be set without a master key. Haven't checked if this also applies to the Class 377. But as far as I've seen this is how most trains work.
When in auto I think, when you've not got the headlights on, the taillights will illuminate automatically when the key is removed from the cab at that end. If you've still got the headlights on, then the taillights won't turn on when you key out. Yeah that's right, you can set the headlights without the master key in. All inserting the master key does is extinguish the taillights at that end when they're in auto mode.