You would edit config.php. Duplicate the connection settings but change them to match the server that is going to hold the user table and assign the connection a different name other than $DB_site (perhaps $DB_user)

Say your forum is the one we are going to use for the user table. We change nothing on yours. You give your friend your database settings, he does what I said above. He then has to edit every php file that uses the user table and change the lines that insert/select/update/delete the user table from $DB_site to $DB_user

Would this be sufficient for what you want?

Your database would also need to have permission to allow your friend's server to connect. It may already allow connections from anyone but you would have to ask your admin, or just try and see if it works.

You then lock your friend into having to make these changes every time he upgrades.