I was 14 when I joined Facebook, in 2008. My dad, who is no dummy, laid out a series of rules for my Facebook use (he had been on the site for a year or so already).
One of those conditions was that he had my password and could look at my behavior on the site at any time. Occasionally I'd post something that he thought was inappropriate, and we'd talk about it, and I'd get embarrassed.
But in retrospect, I'm grateful. Everyone needs guidance as a teenager, and this is especially true for behavior online. Online behavior is at least as permanent as IRL behavior, and the consequences are often more public or serious.
I think it's really important to guide your kids online. Though your 13 year old may not appreciate it at the time, s/he will when s/he's 20 and has only half as much embarrassing teenage material floating around on their Facebook timeline (or Twitter account).