Shooting is completely legal for anyone allowed to legally to operate a firearm. Owning anything full-auto requires said Class III tax stamps/license. This also applies to SBR's, suppressors, and "destructive devices". Then you also need a new stamp for everything you purchase that falls under the whole Class III thing. For example if you were to build a short-barreled, suppressed, full-auto AR w/ M203 launcher you need a tax stamp/registration for all of those parts. The whole 1986 thing applies too; you can build an AR around a registered auto-sear that dates back that far but every other part can be new.
That applies to the average Joe, it may differ for law enforcement because I know some agencies allow you to buy your own AR as long as it conforms to their specific standards.
This is as much as I know, I don't have the desire or need to own something full auto. Just the desire to shoot one, one day. This is mainly because with the paper work involved your local authorities know you have it and I don't need them assuming that I have a armory and drawing attention to myself like that. Also, for the entry price into that "club" (depending on which route you go) you can buy a Barrett and some ammo!
