well, for starters, you can't use the cookie from one domain on another. It undermines the security of cookies.
If you have control over both domains, the thing you can do is set the login script of one domain, pass it's variable to the other domain's login script so that both cookies are set simultaneously.
Then you would do the same thing for the logoff script. One logoff script would call the other logoff script to make sure that both cookies were destroyed at the same time.