Normally, if everything is done correctly, you will get the same result in each browser. Some are more forgiving than others so you might get away with something in one browser that you won't in another. First ensure that you have a session_start command in each module. If you do, then you might want to do a print_r for $_SESSION in the second module right after the session_start() command and see if you have anything in the session variables array. Just to be sure you could do the same thing before leaving the first module.
Also, see the comment below that I found in another post on this topic:
Check the name of the server machine. IE has problems with machine names that contain '-' or '_' - they cannot maintain a session! I've had this problem twice in the past, and it always takes me weeks to figure out, and I'm shocked IE hasn't fixed it.
Don't know which browser versions it applies to but it is worth checking.