I would think fuses but...
Firstly we are talking about 2 different fuses failing at the same time if that is the case. Secondly you mention it began with one side, and then just as soon as that one side began mysteriously working the other side immediately failed. This makes me inclined to think you might want to make sure all of your grounds and plugs are clean, a bad ground can limit current flow and cause not enough to pass to light both at the same time.