You can use mysqldump to export everything to an sql script file. On the other server you can use mysql to import it again. Both are command line tools. Note that users are never exported, you will have to recreate these manually.
What you can do:
Identify all tables in the database mysql which have a field called User. Export those tables with mysqldump. Feed them into the new database. Make sure that you run mysql with admin privileges (usually as user root). Restart the mysql server with the new database after the import. And hope for the best.
If you're in one of those predicaments where cls.__private attributes just aren't enough since they can easily be accessed through inst._cls__private, and you need something a little more secure, here's ...