Determining if there is insulation behind a wall
The easiest way is check behind an electrical outlet in the wall. As a precaution, turn off power to the outlet at your electrical service panel. Then remove the outlet's faceplate. Your outlet will be seated in a metal or plastic electrical box, probably mounted alongside a wall joist. Using a wooden skewer, poke between the front edge of the box and your interior wall board. Make sure to poke outside the box, not inside where the wiring is.

You'll want to use a narrow-diameter skewer, made of wood or plastic so it won't conduct electricity (just in case you bump into a wire). I've used bamboo skewers, which are about 1/8" in diameter and around 8" long, and can be bought in packages of 100 for a couple of bucks.

If you feel you're hitting something soft and pillowy, you've got insulation. If you immediately hit something hard, you're probably bumping into a wall stud -- try the other side, top, or bottom. If you can wiggle the skewer around and touch it to the inside face of the exterior wall -- bad news, no insulation.

If there isn't an outlet on the wall you want to check, you may be able to check by sound. Knock on a wall where you know there is insulation and remember the sound. Knock on a wall where you know there is no insulation and remember that sound. Then compare with the sound your mystery wall makes.

Finally, if all else fails, drill a small hole in an inconspicuous spot and poke your skewer through that. Patch the hole with spackle and a little paint when you're done.