Implementing SSO and authentication
Using Jamroom
Normally for single sign on we suggest using the existing OneAll module
https://www.jamroom.net/the-jamroom-network/networkmarket/47/oneall-social
then they can use the same login on both sites = Single Sign On.
But it sounds like you're looking for something different in getting it to seam like the sites are not different but the same.
My first thought is disable login on the jamroom site, only enable it on the other system then add a listener
Docs: Events and Listeners
https://www.jamroom.net/the-jamroom-network/documentation/module-developer-guide/1011/events-and-listeners
to the jamroom side that picks up on anyone who is not logged in visiting a page.
The listener reaches out to the other system and requests who they are logged in as. Gets their email address and logs in the user with the same email address or creates the jamroom side user/profile with that info if they've not yet got a jamroom side account.
If they're not logged in on the main site and trying to access a jamroom page I'd probably immediately redirect to the login.
Something like that.
You could reach out on each page load to check the user status or if that was too heavy run a listener on the ten_minute_maintenance event to log out users who had logged out at the main site.