After copying across the database make sure that you also make a copy of the database "mysql" which contains the users and the privileges.
From the MySQL documentation:
Do not forget to copy the mysql database because that is where the grant tables are stored. You might have to run commands as the MySQL root user on the new machine until you have the mysql database in place.
After you import the mysql database on the new machine, execute mysqladmin flush-privileges so that the server reloads the grant table information.