I don't think you really want to share session information between two websites at all. From what I can gather from comments, what you're really trying to do is have a user authenticate in one website (give you a username and password which are validated) and then have that "logged in" state transferred to another website which doesn't handle authentication for itself.
What you are describing is the Delegated Authentication model.
In this model, your application hands-off authentication to other systems which it trusts to provide information about users.
There are two well-known protocols which provide this mechanism:
OpenID
This is intended to facilitate users logging in with their own identity providers (Google, Facebook, Microsoft Account). It's a very good choice if you're running a public-facing website, as most users will already have an account they can log in with.
WS-Federation
This is intended to facilitate users logging in with identity providers which are managed by known trusted parties, such as partner organisations.
From version 4.5, the .NET Framework has built-in support for WS-Federation via the Windows Identity Foundation component (and is also available for earlier versions as a separate download). This automates the task of delegating your authentication to an Identity Provider.
It also provides components for you to write your own Identity Provider, should you want to create your own, but you shouldn't have to; you can find various existing implementations to perform this job for you.
The problem you're trying to solve is a very difficult one, especially trying to make it secure enough to be reliable. The good news is that smarter people than you or I have spent years working out very clever ways of doing this. You should use what they have done and not try to cobble together something out of Session state.
In the long-run it's best to let the smarter men do the hard work for you.