Sound suppressors (not to be confused with flash suppressors) do not "silence" a firearm. The .22LR suppressor is the quietest of the suppressors, and pretty much the only one that comes close to those portrayed in films. Centerfire rifle and handgun suppressors drop the sound pressure level down to levels below the hearing damage threshold of 140 dB while using regular supersonic ammo, which is far from silent.
Subsonic ammo reduces the SPL more, but you sacrifice velocity, which translates to shorter effective range.
Yes, they are regulated by the BATF. If your state allows them, you must have the local Sheriff or Chief of Police sign your papers, and also submit fingerprints for an FBI background check, and pay the BATF a one-time $200 tax per suppressor.
EDIT: North Carolina allows suppressor ownership, but the owner must hold an FFL.