If the URLs are different, and I am guessing they are, no session sharing is going to happen easily, I'm afraid. Your choices would depend on the ways you're sharing information between the servers now.
You can send variables in hidden fields in the forms or as part of the URL strings. Obviously this would be public information, even if it is behind HTTPS.
You can set up an API that authenticates requests and exposes the common data. This would afford some extra security if it is behind HTTPS.
The design concept behind sessions is to preserve stateful information from one page of a web site to another. When you're dealing with separate web sites, sessions are almost completely not applicable. Sorry.
Maybe if you describe the application challenges from a "higher level" we can come up with a better solution. Best, ~Ray
