If your son is using an independent application to text, it doesn't affect him at all. If he's using iMessages, then yes. It's recommended that if there are 2 users for 2 separate devices that you have 2 different Apple IDs to zone out the purchases. I know it's pretty common to buy the app on an Apple ID and basically pass the Apple ID around, but if that's the case, make sure you log out of that Apple ID once you're done or you'll end up with iMessages being shared.
If your son's using iMessage, he should be using his own Apple ID since his conversations would bounce back to your device as well.