Well if you keep tabs by going through your child's email, that is your decision and it completely fair to do, after all it is easy to get in trouble on the internet. But to be fair to them at least tell them you are going to do it. When I was about 13 or so I had my first email account and apparently my mom had been reading my messages on a regular basis which I was not aware of. I never had anything bad in there that I was worried about other people seeing but one day I came home to my mother fuming mad. She had an email printed off that was rather explicit and inappropriate. Turns out someone had sent the email to the wrong address but she thought it was sent to me. She had canceled the internet before I even got home, phoned the school and told all my teachers that I had received this email and was to irresponsible to use the computers at school again. She told me that I was to not go near a computer again and if I ever needed to use the internet at school for a project I was to phone her and she would come in and sit beside me while I used it (how mortifying for a teenager).
If she had just calmed down and gave me time I would have gone into my email address on my own checked the message freaked out, blocked the sender, and disabled the account and gotten a new one. After I did that I would have told my mom what had happened and that would have been that. But as it was I ended up being forbidden from using the internet until I was about 16 or 17. So I understand your concern but be sure to give your child a little credit if something does happen.