this reply assumes you're using apache..
session data is stored on the machine which the session is initiated on. The data is normally stored in /tmp. If /tmp isn't readable by apache, then your session data won't be, either.
My main instinct would be to pass the values to the new site with GET or, preferablly, POST, kill the session on the old server, and then start a new session on the https server.
I hope this helps!