Well for one thing, my child won't have their own computer in their room, just my own which everybody in the household will use and that I can monitor. (Added: And yes, I know about kids in school requiring a computer to do their homework sometimes, and one computer will not only meet that demand, but it'll give one more excuse to limit each child's usage on the computer, due to other's requiring the use of it)
Secondly, before I even allow my child to venture on the internet, I will educated the to the full extent of the pros and cons of the internet, what they are at risk of, what other people's intentions might be, etc. If they do not understand something I am explaining, then I will explain that so they can. If they do not wish to take the time to learn what I am going to tell them, then they won't be allowed on the internet until they do, plain and simple.
Thirdly, I would of course be doing very regular checkups on my system and its records of activity. I will, as Mr. L. Wolf mentioned, talk to them. I will not of course do the whole police interrogation approach to the situation, as that usually clams them up on certain details they may think are not approved of. But I will imply that I am interested in knowing who their new/old friends are, how they are doing, what have they been up to, etc. (Getting involved in their life so to speak.)
Forthly, if I found any odd details or information on my system that I am not sure of where it came from, or looks problematic for my child (or myself), I will address the situation ASAP, and confront my child on those details. If they do not give a ligit explination that matches the details I have found, then they will not be permitted to use the system until I do get the right information.
If my child starts to flip shnit over my invading their privacy, I will of course remind them that when they are living in my house, my rules and using my computer, in those regards, they have no privacy.... esspecially when it comes to their own security from other people from the internet. That they should be greatful that I allow them on the internet at all, and if things continue, then I will remove the internet from the household all together, or limit it to only my account on the system which is password protected (And I haven't met anybody yet who could figure out my passwords, as they are virtually random in my head.)
Oh... and if they want that level of privacy in their lives, then they can wait until they grow the hell up, graduate, move out and get their own computer to do as they please with.... until that time, I paid for the damn thing and they will use it accordingly or not at all.
I would of course inform my child that if they have any questions or issues that they would want to bring to my attention from some online experience between them and someone else, or there has been something strange going on, or if they are interested in meeting someone they met online, that they bring it to my attention.
At the basis of every one of these possible situations, they will be approached in a manner in which they are of course not to blame for what has or will occur, and that I will help, with the best of my ability, to give them the proper information so that they can make the right decisions for themselves. If something like the above situation occurs, then I will address that person online in my own way, while I will inform my child of the situation, what occured, what could have occured, and how to avoid the situation again in the future.
I am fully aware that no parent can fully cover every single thing that their children can do in their life, but what I can do is attempt to give them the right information/procedures to protect them when I am not around to do so..... but what occurs on my computer is my business, and until my child get's a bank account and bills to pay online, it will remain that way.