I'm 18 and I've had a computer in my room since I was 15. My parents have trusted me enough, and I don't think there's anything really harmful on the internet to someone with the right upbringing and common sense.
It depends on the kid and the age. Some kids are responsible and others aren't. Age maters too. "tTeen" is only a few years but what a difference between a 13 year old middle school student and an 18 year old high school senior.
My son is 17. I can't think of anything that could actually harm a kid that could come down an Internet connection. I think his skateboard is 100X more dangerous.
My daughter is 10 years old. She uses a computer downstairs where we can watch but so far she seems to be pretty smart about it
That said. I've programmed my wireless router to shut off the Iternet after a certain time at night, Otherwise they don't get up in time for school in the morning. Also, Iknow a little more then the kids do about networks (I've worked in the business for 25+ years) and keep DNS and proxy logfiles. So far nothing stupid is in the logs.