Since the flasher unit doesn't 'know' which side is supposed to flash but woks OK on the left side I would suspect that the indicator lamp earth to the body is the most likely culprit. Front or back? no way of knowing, you will just have to have a look. As the bulb contacts are easiest I would try them first.
If you have a flasher unit for each side of the car then swap them or at least the connections over.Then you will know if the fault is in the flasher unit or the wiring.