Thanks, I missed that one!
So the solution is to use the same database, having each q2a installation with different prefixes (why I didn't think of this!). And having the same prefix for the users table, i. e. "qa_users".
Great to hear that the session sharing for the login works with your subdomains! In my case I have two different domains, so this will not work, to my knowledge. However, I think it's okay to ask my users to login at site B again :)