The Senate is a privileged group. But note that McCarthy didn't directly destroy people's lives. He did things that in other contexts would have been libel and slander, and he did it in the public media. But because he was a Senator, that wasn't technically illegal (IIUC).
OTOH, his smear campaign was used by many other people, some who just didn't want to be smeared themselves, to get people fired, etc. He didn't run into any real opposition until he tackled the Army. Then he lost, because he never had any substantial evidence against anyone.
So people that McCarthy attacked were fired, not because there was some legal requirement that they be fired, but because their employers or supervisors didn't want to find themselves smeared. (And, admittedly, sometimes because of internal politics, and this made an excellent excuse that their enemies took advantage of.) Also, if any company took any federal money, either directly or indirectly, and didn't fire someone McCarthy accused, they faced the strong likelihood of losing those grants. Etc.
Please note that the McCarthy inspired HUAC (House Unamerican Activities Committee) continued up into the 1960's. McCarthy's disgrace didn't mark the end of McCarthyism, merely its weakening. (I've always read HUAC as being a Committee of Representatives that gathered to engage in Unamerican Activities.)