How safe are your kids on the Internet?

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The other day I heard one of the other moms talking about how her little kid surfs the Internet and even chats online. All by herself, in her room. The mom acted like it was perfectly normal behavior, and I just couldn't help but shake my head and wonder. I'm not sure how old her kid was, ten years old, I think. Is this normal to let kids go online and chat with perfect strangers. I've heard so many horror stories in the news where kids give out their addresses to perfect strangers. I just don't think it's safe. I'd never let my own kid do that kind of thing. It bothers me as a mom to see other people doing that. I just have to wonder what these people are thinking when they leave their kids unsupervised like that. Am I just being overly sensitive on the subject? Is it really OK to let kids surf alone on the Internet?