Tables mismatched and not sharing between sites in database.

I attempted to combine three different databases from three different sites into one database with some tables shared and others prefixed for their respective sites (keeping backups of each, of course). Not all of the shared tables are sharing like I understand that they are supposed to. My log in status is not the same in all sites despite sharing the sessions table, and user settings set in one site don't show up in another despite sharing the users table. Also, the Schema module says that for several of the tables "...the schema and the database are different," whatever that means.

How can I fix the database that I've created or recreate it the right way? What should I be reading to understand this problem better?

Thank you.