They are legal, but also heavily regulated. Its not something you can go into a gun shop, purchase, and walk out with the same day. It takes time, generally months to obtain. There are 2 ways to legally obtain a suppressor in the USA. One way, you find a gun shop that is licensed to sell suppressors. You purchase your suppressor. The dealer gives you the paperwork you need to complete the approval. It includes fingerprint cards, and forms that include the manufacturers info, model, caliber, and serial number. You go get fingerprinted at the police station, and pay for a background check. You go get 2 passport style pictures taken. After the background check results come back, you then need to visit your local Chief Law Enforcement Officer. It could be the Sheriff or Police Chief, or other high ranking Law Officer. He will conduct an interview with you, and if he decides everything is in order, and you dont have any criminal issues in your background history, he signs your forms stating he sees no reason why you should be denied. You then send all the paperwork, pictures, and fingerprint cards to the Bureau of Alcohol, Tobacco, Firearms and Explosives, along with a $200 check. They review all the info, do a second background check, and several months later(4-12 months) they make a determination. If approved, they send the approved paperwork back to your dealer(who has held your suppressor in his possession). The dealer then contacts you to come pick up your paperwork, and suppressor.
There is another, different path to obtain a suppressor, which involves setting up a Trust, but I dont know how that actually works, so I wont offer an explanation.