The two sides of the map light are configured in parallel.

But, when I install LED festoon bulbs in each side, only one side lights up.
I put them both in and one didn't work, I thought it was burnt out or something.
So I took them both out and only put in the one on the side that wasn't working and it worked, but only one side works at a time. I can't turn both switches on and have both lights on, only one.

Here is how it's wired from behind:
any ideas? do I need a resistor?

 

Resistors limit current, so that's not the problem. Do you have the polarity correct when you're installing them? LED's have a positive and negative side. My map and dome lights are controlled from a switching ground. Is that how yours is connected?It probably is, but I can't tell from the photo.

EDIT: Can you take a shot from directly above the rear of the housing and also take a photo of the LED's you're using?