They are legal. Well, it varies state by state, but at the federal level you apply for your Form 1 or Form 4, pay your $200 tax, and then wait 6-8 months for background check and (hopefully) approval. Then proceed to the store to pick up your can (or if self-manufacturing, you can then machine it yourself and engrave your serial #).
In my state of Washington, until not too long ago you could own a suppressor but not use it. They finally remedied that weird catch-22 a couple years back so you actually get to use them now.
States have varying laws... I think it was something like 42-44 states, they're legal to own and use. Some states have other restrictions on other NFA items. WA for instance doesn't allow full auto. Other controlled items are short barreled shotguns and rifles (SBS/SBR). Those are also on a state-by-state basis. And same deal... you apply for it, pay the tax, and then wait. When you're approved - congrats, you get to take a hacksaw to your barrel and do the engraving.
The other prohibited item (am I forgetting any?) is the "any other weapon" which is where the ATF gets to be a little fuzzy. It might include guns disguised as cell phones, or spring-loaded blades (think switchblades), or whatever. The AOW category is where I would guess they'll throw bump-fire stocks... any rifle fitted with one would be classified an AOW and require the stamp. If I had to guess.
The evil genius of it is that the ATF could, in theory, do that now, without congress, because they've been given pretty wide latitude to regulate stuff like that. It could be challenged, but the ATF has regulated other stuff in the past that used to be legal and the ruling stuck, so...
Still, they are apparently quite difficult to own legally because of the taxes, the forms, and the wait.