User privileges in MySQL are host specific.
so you must grant privleges to the remote user. If for isntance root shall be able to log on from everywhere and have same privileges as 'root'@'localhost' has the SQL is
grant all on *.* to 'root'@'%' with grant option;
On an example where 'billy' may only select from a single database and connect from a certain group of ip's.
grant select on *.* to 'billy'@'10.0.0.%';
http://mysql.com/doc/refman/5.0/en/grant.html