I\'ve searched for an answer to this and all I can seem to find are questions asking whether it is better to use multiple databases or multiple tables in a single database.
You can grant access to different database to different user using GRANT in MySQL.
https://dev.mysql.com/doc/refman/5.1/en/grant.html has the information you need.
The most simple you can do is
CREATE DATABASE db_for_user_a
CREATE DATABASE db_for_user_b
GRANT ALL PRIVILEGES ON db_for_user_a.* TO user_a IDENTIFIED BY 'user_a_s_password'
GRANT ALL PRIVILEGES ON db_for_user_b.* TO user_a IDENTIFIED BY 'user_b_s_password'