It depends if the other domain is on the same machine as the original one. If it is, you can pass the session_id to the other domain and it can pick up the session, for example, I have two pages, testing.php and testing2.php, and two separate domains on the same server:
testing.php:
<? session_start();
$name = "testing";
session_register("name");?>
<?=session_id();?>
<br>
<?= $name?>
<br>
<a href=http://some.other.server.com/testing2.php?sess=<?=session_id();?>>goto another server</a>
testing2.php:
<?
session_id($sess);
session_start();
?>
<br>
<br>
<?=$name?>
and $name will be passed to the new server because it just picks up the same session from the first by the session id. It is apparently important to set the session_id before calling session_start (see http://www.php.net/manual/en/function.session-id.php).
Now if they are on two separate boxes, then it depends on how you are ultimately storing the information. If the information is being stored in a database that both machines have access to, pass a unique database identifier and repopulate the session from the database on the new machine. If not, you will have to pass (through a form post, with hidden elements or the like) all of the information needed to rebuild the session on to the new domain.
Make sense?