The House Un-American Activities Committee (which usually goes by the acronym HUAC, btw) was a committee of the House of Representatives and as such could not prosecute anyone (Congress has no power to prosecute). It was solely an investigative committee. However, many people who were subpoenaed to testify refused to answer questions on the grounds that doing so would violate their 5th Amendment rights against self-incrimination. Some, such as the late folk singer Pete Seeger and the Hollywood Ten, were later prosecuted for Contempt of Congress for their refusal to testify. They weren't prosecuted for being a communist, per se, but instead for refusing to answer Congress's questions about whether they might be a communist or had connections to the Communist Party.

Others were prosecuted under the Smith Act, which made it a crime to advocate for the overthrow of the US government by force or violence. The Supreme Court initially upheld the Smith Act as not violating the First Amendment, but by the late 50s started narrowing its scope. In 1957, the Court held in Yates v. United States (a case involving the prosecution of members of the Communist Party of the United States) that the Smith Act only prohibited actively advocating for people to overthrow the government and that the First Amendment protected the people's right to teach the idea of violent overthrow of the government as an abstract idea. So, again, they weren't prosecuted for being Communists per se, but instead because they advocated for the violent overthrow of the government (which just happened to be a tenet of their Communist ideology).