if you use a database, though, the other site will also need access to the database.
you could also try to serialize($_SESSION) and then put the result in a cookie on the user's browser and then have the other site retrieve the cookie and unserialize it.
IMPORTANT: if the session data is sensitive, it will be at risk in a cookie.
yet another way:
serialize the $_session data, put it in a form input field, and have that file POST the data to a script on the other site that knows how to handle the posting.