There are many reasons and you have to isolate them one at the time. First, to make it easy on yourself, you should have a volt ohm meter or at least one of those cheap test lights with a little alligator clip on the end of a short wire and a point on the probe end. Start at the plug on the vehicle with the trailer unplugged. Those plugs get cruddy at times causing a bad connection so if it's dirty you just as soon clean it before you start. It helps to have someone sitting in the vehicle also. Clip the alligator clip to the frame or bumper for a good ground. The with the running lights on, test each of the plugs holes to see if you have power. On a 4 part flat plug, the stud is the ground and only one of the 3 holes should have power, and since you say you have one trailer light working you should find it easy. You might have to scrape the probe in the hole to make a good connection. Once you know which is the running light the other 2 are your brake and signals, one for each side. With the ignition switch on put the left signal on and check the plug for power in the 2 hole other that the one you have already determined was the running light. If all is right with the vehicles lights and wiring one should have power going on and off. Switch the signal light switch to the other side and the last hole should have power going on and off. If that all checks out then the problem is with the trailer. Plug the trailer in and disconnect it a couple of time to clean and make a good connection. Then go to the back of the trailer.
Even so called waterproof lights seem to get water in them but if it's easy to do take the lens off of the trailer lights. You should remove the bulbs and check them visually to see if both filaments are intact. If you find either of the two filaments broken you need a new bulb. If they look good you might have to clean the contacts on the bottom of the bulb by scraping them with you pocket knife or a piece of sandpaper. Once you think the bulbs are OK you can test the bulb socket for power. Put your running lights on again without the signals on and test the sockets for power. One of the two contacts should have power. The other should have power when you put the signal on. Test both sides. If they have power your cleaned bulbs should work or you need new ones. If there is no power in the sockets you could have a bad ground and need to fix that before you can go any further. If necessary you can use a battery jumper cable from the vehicle frame to the trailer frame for a ground. Then test again for power at the sockets. If you have power then your problem in your ground. If you still don't have power in the sockets you probably have broken wires or your connection to you lights is bad or corroded. You can check those by taking loose the wires from the lights and check them for power. If you have power on the wires, clean them and reinstall them. If you don't have power, you probably have a break in the wire somewhere between the plug and the end of the wire.
To get it working is no big feat. Just a process of elimination. Oh, and did I say you need a good ground.
"gene"