I would strongly advise that you don't rely on mods to move them as they see fit: this suggests that you'd rely on the mods to move a file from private into being shared.
That puts liability with the mods. You don't really want that.
My recommendation: have a published policy for how to complain if you are a copyright holder and what actions the site will take.
"If you are a copyright holder and you have found material that you wish to have removed, please notify us. We will immediately put the item(s) in question unavailable awaiting proof of copyright status. This can be provided in the form of..."
If you see what I mean. The onus is with the copyright holder to prove infringement in these sorts of things, and they will be the exception rather than the rule...