You can't access the same "session" from completely different servers.
From what you describe you just need to use the same login, e.g. you don't want to provide your credentials again. This is, as dove answered, done with a Single Sign On solution. The Wikipedia article contains a lot of useful links.
Usually you use a third party service (that you also can provide yourself, you don't need anybody for this) and some redirects between the different servers in order to get some kind of certification of the identity of the user.
Basically OpenID, e.g. used here on stackoverflow, is a solution to the same problem, though you usually don't have implicit login - you need to explicitly log in.
If you control both participating servers as well as the authenticating "third party" (which may in fact be part of one of the two servers) you should be able to provide some implicit login easily.
Be aware of possible privacy issues though, if the sites don't obviously belong together. People might feel alienated if you proactively shared their identity if they don't recognize that both servers belong to the same entity.