The session is stored on your server. You can't really store it anywhere else, but you can use cookies to either store the values on the user's computer, or I believe you can store the session_id in a cookie, and then load that session from their computer. I'm sure there's probably some kind of security measures you need to take, but take a look at:
http://us.php.net/manual/en/function.session-id.php
http://us.php.net/manual/en/function.session-encode.php
session_encode encodes the session as a string. You can probably store that string as a cookie and then decode it later.. session_id sets/gets the session_id. You can store this in a cookie and load it if the cookie is set.