One only has to be 13 years of age to sign up to facebook, although I'm sure younger ones get on there anyway. When playing a random game, I am quite careful. I also turn off the chat function, as i have no interest in a conversation, just a good game of scrabble. When I play against friends (online or at home) I will play any legitimate word that suits my ojective (scoring points, blocking etc).
Although I agree that parents should monitor their childs internet usage, I'd imagine that if they saw them playing scrabble they'd be quite content and not watch over them, so no, you are not being overly sensitive.