zoomiest My hosting support guys have asked me: "Exactly what permissions are required?"
I find it very strange that your hosting asks about this. PHP on any hosting service should have WRITE access to the files it can access.
Permissions depends on if you have different users, for example FTP-user uploaded the file, and then the PHP WWW-user is a different user with different permissions. Normally, there should be one user on a shared web hosting service (same user for www, ftp, file system) so that there are no problems with permissions. Your hosting should know this. If the index.php is owned by FTP-user and the file systems is owned by another user, then you will need to assign permissions appropriately for the file system so that users can write within other users. This doesn't just go for Files Gallery, this goes for any PHP application that wants to write into the file system.
As noted, if you have same "user", this would never be a problem.