I used to be a journalist at the UK charity The Fostering Network, and one of the final pieces I wrote there touched on exactly this question - how to keep young people safe on the net without stiflling the fun of it all.
When I looked into it, the CEOP was still in its formation stages, and it was hoped that in itself it would be a great resource for parents (and indeed foster carers) in the future. The headline horror stories were already prevalent - tales of men grooming young people on the net for intrusion, bombardement with obscene images, and real-world abuse.
Two things became apparent. Net access might be fun, but it shouldn't be available to young people on a blanket basis. If the child is young, they should be supervised during their net-time, in a way that makes that net-time as much of a 'together' activity as anything else - from sport to TV to constructive play. This will not only turn what could be an isolating experience into a bonding one, but will reinforce the idea that the parent is always there to help with any future problems or questionable moments that they have on the net. Essentially, the net is like a vast, sprawling, invisible city - if you wouldn't be comfortable leaving the child alone to explore that real-world environment, don't leave them in the online equivalent. And secondly, of course as children grow older they need to be trusted, to explore on their own, so earlier solutions like having scheduled family net-time or having the computer in a communal room will no longer be effective.
There are technological solutions that can be used of course - net nanny programmes, administrator logs etc. But ultimately these are just a safety net, and it should be remembered that young people are in all likelihood more technologically minded than their parents. So the key to keeping them safe is information and building a relationship of trust. Make sure they know that not everybody is what they say they are, and teach them the importance of not divulging too much real-world information, especially addresses, phone numbers etc. And talk to them about the net on their terms. Let them know that you're there for them if anyone starts asking inappropriate questions or sharing inappropriate information - information that makes them uncomfortable. Let them know that there's no judgement coming from you about anything they might be sharing with someone else, that you understand they might need to share all sorts of things with their online friends, and that there'll be no comeback if they need to show you all that, because keeping them safe and comfortable is more important to you. Similarly, make them aware of the other resources available to them - the Childline number etc - and again, let them know they can use it with no comeback from you if they need to.
Keeping children safe on the net is essentially the same discipline as keeping them safe and not smothered in the real world. Introduce them gently, and hold their hand for as long as you feel it's necessary, then tell them the dangers, teach them what not to divulge to strangers, invest your trust in them, and make sure they know that if they end up in dangerous situations, you'll be there for them any time, just as you would be if they called you late and alone on a Friday night. It might not be the coolest thing in the world to get your parents involved, but make sure they know it's cooler to do that than it is to be alone and vulnerable in tha big net world.