How do you access the rear license plate light bulbs in a 1995 miata mx-5?

Which bulb are you needing to replace? Most of the snaps in the Miata have a center piece that pops out first using a small screwdriver then the whole snap can be slide out of place.

There are two clear plastic covers on each side of the rear license plate with bulbs in them (I guess) because one side is lit, but the other is not.

You are correct then. The panel inside the truck does come out by removing the plastic snaps. Lift the center of each with a small screwdriver. The center will "pop" out and then the whole snap will come out of the hole. You should be able to wiggle them loose with your fingers once the center is lifted. The snaps are in a plastic/rubber hole for grip so it can take a little wiggling to free them the first time. If needed, use a pair of needlenose pliers to grab the stem of the snap carefully and wiggle free. After the panel is removed, you'll see the bulb socket. That will twist out (1/4 or 1/2 turn). Can't remember the bulb type, though. It is a common type available at auto parts stores. Let me know if I can help further!

Thank you so much; I'll see what success I have. I was just afraid to try to pull on those snaps for fear of breaking them or not being able to get them back in.....reminds me of some door panels I haven't had good luck with in the past. :o)