Documentation 4.12.10 states:
When enabled, both the admin and the uploading user can edit and delete files that were uploaded to a public gallery. Of course the user must have permissions to upload to the public gallery in the first place (see the groups control panel) before this option will have any effect.
..........
Does this mean that the uploading user can independently edit or delete files without approval from the admin?
Yeah. It gives them the same rights over their public files as they have over their user gallery files.