As far as I know, crossing sessions between sub-domains is fine, but it won’t carry over to a whole new domain. To do that you need some sort of centralized data method, or an API.
Database method: you will have to create a remote MySQL data access so that domain2.com can access the database on domain1.com. When a log-in is performed, not only should it create a new session, but a unique log-in token (with an expiry time) should be put into the mysql database. Now, for every link that goes from domain1.com to domain2.com, you should add a $_GET variable that contains a randomly generated session id (md5 hash will do). domain2.com, upon receiving the visitor, will take the $_GET variable, run it through the MySQL database to find the login token, and if there is a match, consider that user to be logged on (and perhaps embed a $_COOKIE as well to store the login data). This will make the log-in transferrable between two completely different domains.
API method: you need to create an API method, so that domain1.com can respond to an external request from authorized domains to retrieve the login token upon a user being forwarded. This method will also require that all links going from domain1.com to domain2.com to be appended with a $_GET variable to pass the unique session hash. Then upon receiving the visitor, domain2.com will do a curl() request to domain1.com/userapi.php (or whatever you call the file) and the variables should be tested against what’s in the database.
This is the best I can explain it.. to write this out in code is a significant piece of work so I cannot commit. But judging by your code, you have a very good understanding of PHP so I’m confident you will pull this off!
Good luck mate.