How different are the urls? Do they share the same domain?
Are they multi site installations based on folder names?
The only thing I can think of is if they share the same domain and they are configured as multi site installations by folder then the siteguid is used in the authentication cookie and since you copied the database the siteguid would be the same possibly causing some kind of cookie conflict. But if they are not the same domain I see no way that they could impact each other.
You might try varying the name property on the <forms element in web.config so they are not the same accross those sites.
In any case without steps to produce the problem it is not something I can investigate. It is certanly not a problem I've ever seen or heard of before.
Hope that helps,