There is a lot of confusion about the legality of hunting with suppressors on the Internet. Part of this comes from the fact that many people don't realize that the world is not the same everywhere you go. The United States is comprised of 50 states and they all have their own laws, so what it perfectly acceptable in one state may be punishable by a fine and jail time in another.
These days, the ownership and use of suppressors, also known as silencers, is getting very popular, so much so that several states this year have amended their laws to allow hunting with suppressors. The use and ownership of suppressors is really not difficult. You fill out a Form, get some fingerprints and pictures, get a CLEO sign off and send 200 bucks to the BATFE. The only thing hard about is waiting for the BATFE to approve the paperwork so that you can pick up your suppressor when the paperwork comes in.
Of course, it must be legal to own in the state you reside in. Some states allow ownership of them but not hunting, others allow ownership of them and hunting.
There are no downsides to hunting with suppressors really, in fact, its a good thing to do. In many cases a properly mounted suppressor will actually enhance the accuracy of the bullet. This is because the suppressor( which we'll call a "can" from here on out) strips the gases that normally overtake the bullet and destabilize it by slamming into the base. The bullet due to its shape, eventually stabilizes itself, but without the gasses acting upon it, it flies much truer and thus is more accurate. Also, recoil is reduced considerably. Another benefit to that is that the reduction of sound also tends to help "flinching" and anticipation of the shot that can cause one to pull the shot off target.
The best thing about hunting with a can is that because the shot is alot quieter, you will see more game than you ever saw before because you aren't alerting everything within a two mile radius when they hear a shot. An example of this would be when squirrel hunting. When using a can it is common to be able to shoot several squirrels out of one tree...to them the shot doesn't sound much different than an acorn falling through the leaves, they barely notice it.
I probably don't need to mention the obvious, but I will. It is so much easier on the ears that shooting becomes a pleasurable experience. You can talk to your buddies in normal tones and when one of you shoot, it wont ring your ears. It really does protect the hearing.
As of right now, the it is legal to hunt with silencers in the following states. According to Gemtech, one of the leading silencer manufacturers,here is the list from their website:
Montana (non game animals only)
Its getting better and more states are considering loosening the restrictions on suppressors. As always, check your local laws and regulations to stay current because the laws are constantly changing.