I've had this problem in various versions of Opera 9, and now that I've installed 10, it's still there, so I was like "no way, let's see what we've got here". So I've found why it doesn't work.
Press CTRL+F12 To go to Preferences. Choose the Advanced tab, and then the Cookies group of options. Select "Accept Cookies". Log in to YouTube, choosing to stay signed in. Now go back to Preferences and switch back to "Accept Cookies only from the site I visit", since this must have been selected, causing the problem (you should switch back because it really is a good option for some extra security).
Also, I've temporarily re-enabled the sending of referrer information, but I'm not sure if it was necessary. I keep this disabled as well, and most sites work properly like that. (If they don't, that's probably bad programming, because sites shouldn't rely too much on user-submitted data). If you don't know what it means, though, don't change it, just leave it how it is now.
Anyway, the actual problem is YouTube trying to store cookies from other servers, and that's sort of a no-no for user security. I hope they change that in the future.
Hope this works, cheers!