Depends. Some DD's are listed under the NFA of '34 and as such had to be registered pre-84. Unless you are a government agency, newer Class III toys and some DD's are strictly off limits. The Master Key and the AA-12 are both NFA DD's.
Being NFA doesn't completely eliminate ownership. I can go buy new DDs now, just have to file paperwork and pay $200.
I just bought a brand new, way-post-'84 NFA SBR. Hardly "strictly off limits". (Where'd you get the '84 date?)
The only thing "strictly off limits" is post-'86 machineguns. Anything else is legal with proper paperwork; if hard to get, either is a state restriction (all NFA banned in HI) or simply an inadequate market (I'd like to buy a new XM-177 howitzer, but don't see any sellers).