The bottom line with avatars is the site's safety. By preventing the site from hosting and displaying content that does not belong to the site, we do not break any image copyright laws and pretty much avoid the whole iffy messy junk on if there's any laws with that or not. Personally, I beleive in free information exchange when it comes to art, but I don't make the laws, I follow them.
I know avatars can be precious to people, and some people spend years looking for just the right one. Or when someone draws them an avatar, it's one of the best online gifts they can receive, and I don't want to take away the right to use that gift. But, before anyone uploads anything to this site, we really need them to give us the rights to host and display it, often even modify it (saving an image as another format, for example). And if no one has directly given you the rights yourself and the right to carry on those rights, then we simply cannot remain a law-abiding site.
I really try not to be strict. Actually, what I do is I look for suspicious avatars, check the file name, and google it. If I get a crapload of results, then we have to play it safe and say you can't use that avatar. I don't do it for everyone's, especially when it's obviously a hand-drawn dragon I haven't seen before or have any memory of.
Yes, obviously I'm missing a few. And you may even notice that I don't keep the rules TOO strict on the site's personal rules like avatar dimensions and filesize. I let a few squeeze by. My rule of thumb is that if some law-enforcing person came to the site and couldn't easily find out who's using an illegal avatar, then we're fine. All I have to do is do that job before s/he does, and we should remain safe.
One personal reason I wanted this site rule is because when I come to a dragon forum, I like to see hand-drawn dragon avatars all along the thread. I don't like to see stupid smilies, crap pulled off the net, or such. I like to think that the avatar represents in part who the person actually is, in dragon form or not. And I identify people around here better by their avatar than their names, so you can see how that could be kinda important for the feel of our forums as well.
I hope many of you feel the same way and have enough respect for the site as not to place it in a state where it can be threatened by law enforcement.