(For Dutch/European people mostly) A lawyer could provide more precise details but in general, when you write something original and with a personal mark (signature) then your comments would be protected by copyright. However, many responses here are just facts to help someone to solve a problem and facts cannot be copyrighted! However, comments could break certain laws, especially when you start to put a hyper link in your replies. Most links would still be okay, though. But a link to a racist site could be illegal in Europe. Or a link to illegal software. Or a trademark violation when the link claims to point to HP, yet the URL points to Dell... The author is the one responsible for these remarks, the site owners just have to remove those "illegal" remarks or become partially responsible themselves.
If comments would be owned by the site moderators, they could end up in a lot of legal problems. Basically, in that case a visitor could post a link to illegal software and the site owners would have to face legal consequences. They would not be protected by the safe harbour principle, since they own the comments! So smart forum-owners will never ever want to own copyrights to any comments.
Another interesting part is that posts here fall under the cc-wiki license. Basically, that's the main rule here when you start participating. This is a rule that you accepted when you joined this site, and it can be enforced simply because these rules are a visible part of every page here! However, changing those rules would not be allowed in Europe, so if the owners discover an error which they want to fix, they do need to notify the members about this change, allowing members to resign from this site!
But what effect would resigning have for your posts? You could revoke the license on your comments but that would only have effect from that moment on.
And what does the cc-wiki actually mean? Well, it means that when someone is copying your comment, they will have to make it clear whom wrote the original text. (Attribution) Also, if your text becomes part of a bigger piece of text, that bigger piece of text would require at least the same CC license. But hey, I could publish a book with all the questions and answers from this site and sell it, without having to share a dime with any other visitor here! The book would just need proper attribution and others would be free to make copies of this book and share it for free in digital form. (Or Xerox the whole book and give it away for free on dead trees.)
And what if you don't agree with this? What if you want to comment, but you don't want your answer to be part of this license? Well, you could always put the answer on your own website and add a link here to your site. However, you cannot copy the question to your site because your site would then fall under the cc-wiki license. Then again, if the question is very simple and didn't need much creativity to write, you can use the question without being infected by cc-wiki. Why? Because the text still needs to be creative. Just asking if anyone knows a good Paint application is not creative. Asking for the risks of upgrading from Windows 2000 to Windows 7 isn't creative either. Telling someone how to install the latest driver for their network card is informative but maybe not creative enough to be licensable... (You're just mentioning facts.) Even the description of a film plot on the back of the DVD you just bought might not fall under any license since it just mentions what's happening in the movie. (Which are facts.) However, showing a 10-seconds preview from the same movie or even from the trailer might require a license, though. Unless you're describing that scene as part of a bigger article, in which case it becomes part of a "quote". You can't even use a screenshot from a movie, if it's just meant as decoration. An Ewok image which I saw somewhere on these sites might already be a violation. Then again, it was used as part of a document, referring to the movie and Ewoks in general, making it perfectly legal again.
Basically, licenses aren't black and white. They're just a whole lots of shades of grey. Some dark, some light. But just remember: if a forum owner claims ownership of every message in his forums, he also becomes legally responsible for all those responses. A smart owner would never even consider that kind of responsibility. (But they might add a restriction to the license that you can use here.)