U can share session in between ASP and ASP.net through cookies. If the session state is stored in a common format in SQL Server, the session state can be accessible by both classic ASP and ASP.NET.
When a user makes a request to the Web application, the user will be issued a unique cookie to identify the session. On subsequent request, the browser will send the unique cookie back to the server to identify the session. Before the requested Web page is loaded, a custom object will reload the user session data from SQL Server using the unique cookie. The session state is accessible in the Web page through the custom object. After the Web request is finished, the session data will be persisted back to the SQL Server as the request terminates.
I hope this might help u.