A bad ground can do crazy things with trailer lights. It may be that your ground wire is just good enough to handle one side of lights. Undo the ground wire on the trailer and clean the spot to make sure it is touching clean metal for the best ground. Also, each trailer light assembly should be grounded. If your lights are mounted to metal, they are generally grounded when they are screwed in. If the lights are mounted to PVC or wood they won't be grounded the same way, you will need to check to see that they are good and grounded to metal. Also, if you are using a tilt trailer, it could be that the ground isn't reaching the back.
Hope this helps.