Well, the problem that I've not considered yet (at least, one of many I've not considered) ... are these servers actually on the same system?
Because, if they're not, then the same session data won't be there.
Of course, you could send a flag to the second server, "hey, he logged in over here and it's cool!" ....
but that would be uber-*insecure*, I'd think, wouldn't it?
Now, if the "secure server" and the "log in server" (for lack of better terms) are actually the same box, then it's not so much of an issue. They should share the same session repository (often it's /tmp or something similar....), and you simply need to call [man]session_id/man prior to calling [man]session_start/man on the receiving server's script.
If they are different boxen, your task is more difficult. Sometimes in a "server farm", you have a shared database, and many applications will enact shared sessions in the database itself.
Otherwise, you're left with simply telling server B that the client successfully logged in at server A and stick a name tag on him. I'm sure there's creative ways to do that, but that's kinda outside of the scope of the volunteering we do around here at PHPBuilder, IMHO.... (gotta eat, y'know?) 😉