[Tfng]

Amateur hunter a simple thing to do would be to remove the lense on the right hand light and see if there's anything obvious going on in there.

Your brake lights and signal lights use the same wire and bulb filament so we might be safe to assume the wires for those to that light are ok.

We don't know if the wire is good for the tail light circuit.

Inspect the light bulb for being broken (they shatter if you put hot bulbs in water) I'm doubting it since the signal still works.

Inspect the bulb and make sure both filaments are good and they are not touching each other.

Inspect the ground off that bulb socket. The bulb grounds through the outer metal part of the socket if it's rusty it may not get a good ground. Sometimes there is a ground on the mounting plate. It's possible (even likely) there's no ground at all and it's using the trailer frame.

If you look into the bulb socket you'll see two lead terminals, that's where the power comes into the socket. The plate those terminals are in must be able to slide a bit in the socket to maintain tension on the bulb terminals.

There's a few things to look at without knowing precisely how it's wired.

A test light is a handy tool and easy to use. Well worth the 20 bucks if you're going to be pulling trailers. Trailer wiring can be a never ending battle it seems. Most factory wiring jobs are very poor.