@dielan Do it. Even if they use pseudonyms, they can post something that could link back to them.
Look at it this way, unless your children are home-schooled your kids MAY end up in a news story on the school newspaper which may end up online, and they often use the kids real names without your permission. Now, while your kids may not actually use their real names online, they might drop hints (like something unique about their school even without giving it away; let's say that the principle's office is literally in the hallway), and people can make searches from their. If enough information is posted, they can pin down the exact school, and even the exact person.
Once, they are on their own, they can do what they want.
@dielan You also have no control of what their friends would post about them. I had my first name revealed on a chat room I ran years ago, because someone who knew my personally mentioned me by my real name. Granted, it was only my first name, but still.