Many people seem confused about the concept of freedom of speech. The First Amendment literally constrains only congress. Here it is:
Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the Government for a redress of grievances.
The courts generally understand that this also constrains the executive, since there cannot in principle be a law that gives the executive (i.e. the president) the power to violate these prohibitions. Subsequently, the 14th Amendment extended these constraints to the states.
The courts have also concluded that this should not be taken too literally. Many crimes are committed by means of speech. Frauds and cons are not legal, nor are coercive threats and dangerous deceptions. (The classic "yelling 'Fire' in a crowded theater.") Although the amendment was obviously intended to protect political speech, the courts have, in my view unfortunately, extended it to commercial speech which is why it is so difficult to regulate deceptive advertising. Individuals who are defamed can sue, although the courts have made it difficult to sue "public figures," as probably it should be. People will make mistakes, but only willful or recklessly negligent defamation of public figures is actionable.
So what's the current brouhaha about "cancel culture" or claims that "the Left" wants to suppress speech? First of all, let's be clear that this has nothing to do with the First Amendment or anybody's constitutional or legal rights. The First Amendment constrains government, not any private entity. You have a First Amendment right to say what you want to say, within the above specified constraints, but you don't have a First Amendment right to have your op-ed published in the New York Times. Publications, broadcasts, web sites -- editors and publishers decide what they want to present to readers and viewers. Many have an overt political point of view, others protect some version of what they consider to be truthful or important enough to publish, and that's their choice. I don't expect to get published in Breitbart News or the Wall Street Journal op-ed page and I don't claim I'm being censored, although I suppose in a sense I am. But that's their right.
So if people complain about an editorial decision by, say, the Times, and the editor-in-chief fires the editorial page editor because of it, that's their perfect right. Objecting to what somebody says or writes is not censorship. Saying that I'm not allowed to object, on the contrary, is to call for censorship.
Then there is the question of academic freedom. There is no academic freedom at Liberty University, and I don't hear people on the right complaining about that because they don't think there should be. Liberty University exists to promote an ideology. But what about a hypothetical Ivy League university, let's call it Fuchsia U? The biology department does not grant its professors the freedom to teach creationism because that lies outside their definition of biology. The medical school does not allow teaching of homeopathy, and the sociology department does not teach racist theories, for the straightforward reason that they do not believe these theories to be true.
What about a professor of any subject who espouses racist or misogynistic theories? This professor is disqualified because she or he cannot provide a supportive educational experience to many students, or fairly evaluate any of them. Non-discrimination, equity and inclusion are core values at Fuchsia, and it is the right of the trustees, the deans, and the faculty to promulgate and protect those values. However, the speech of these hypothetical racists is not suppressed. They are free to say whatever they want, they just aren't free to say it while they are professors at Fuchsia U. And any and all employers have the same right to protect their employees and customers from discriminatory or harassing behavior by employees, in fact they are obligated to do so under federal law.
What about a student group that wants to invite a racist speaker? Again, as with the New York Times, the university is under no obligation to provide university sponsorship, or even a venue, to a speaker it finds objectionable. These are finite resources to be deployed as the institution sees fit.
So no, people on "The Left" do not want to censor or suppress speech. They just don't want to be forced to promote speech they disagree with, which is everybody's right and which, in fact, the courts have found also to be inherent in the First Amendment. So go whine someplace else.
The Proud Boys don't shout people down, they threaten to kill them with semiautomatic weapons. But that's their Second Amendment right.