You should remove the housing and look at the switches to see if the switch is sticking on the problem side. Consider swapping the working side switch with the nonworking side one and see if the problem follows.
According to page 22-256 in the repair manual (find in sticky thread), there are two screws - one on each side just to the outside of the bulbs - that you need to drop the housing.
Here is what I remember when doing this for the front map lights on my 2011 since the driver side map light stopped working. The one screw touching metal may be the ground but it holds the housing along with a second screw. You want to drop the housing down so you can access the switches. I would disconnect the cable behind the housing while you are moving the switches to avoid live power issues since these lights get unswitched power from battery (which is why they can run your battery down if left on). You can leave the housing plugged in and dangling from cable while you test switches but only touch the plastic parts, not metal in the housing as those route the power.
You know the housing unit has power since one side works - so it is either the bulb, switch or electrical connections in the housing itself. Since there are two bulbs and two switches, I suggest moving the working bulb and switch to the nonworking side one at a time to isolate the problem part. You may want to check and clean the contacts for the switches after you pull them off if there is any sign of buildup. If you can't get the working switch/bulb to work on the nonworking side, then the housing could be the issue. The thin metal that runs through the housing on each side is the circuit for the power. If you see a crack in the plastic housing or thin metal circuit then that could be your problem . I found such a crack on my driver side map light thanks to a certain driver who aggressively presses on the map light. This also cracked the thin metal circuit on the one side so now the circuit is can't be completed.