According to the state action doctrine, only government entities can violate the First Amendment. Twitter, Facebook, etc, are not government entities. They don’t violate anyone’s constitutional rights when they take down posts or remove their accounts. That does not mean that Twitter, FB, etc, aren’t censoring speech.
Some worry that Twitter, Facebook, etc are monopolies, violating anti-trust laws and that they thus ought to be regulated as common carriers. This strikes me as pretty obviously mistaken. Not only are they competitors, but others have sought to compete with them (MeWe, Parler, etc) and nothing rules out others trying to do so in the future. Some also worry that Twitter, Facebook, etc might act in ways meant to curry favor with the Federal government and that if that is true, since its plausible that federal regulators know this and might thus signal their desires to these firms, really Twitter, Facebook, etc are agents of the state and so they can, after all, violate individual constitutional rights. This also strikes me as pretty implausible, both legally and morally. Whatever control would be present would be pretty tenuous. If it weren’t, it seems unlikely that Twitter would have closed President Trump’s account.
When Twitter bans President Trump from its platform, it prevents him from speaking to a certain audience, limiting his speech. It does not thereby successfully prevent him from speaking to everyone; he has other avenues of communication. Of course, if the government censors someone, they also will typically have other avenues of speech. Consider the Comstock Act of 1873; it made it illegal to send certain “lascivious” material through the mail. Those wishing to share (speak about) those materials with others, could still do so—for example, by walking to others and talking to them directly. More generally, any governmental act meant to silence someone will close some avenues of communication while leaving others open. The fact that a social media company only closes some avenues of communication to (i.e, only partially silences) someone it bans from its platform is no different than what government does. If the latter is censorship, so is the former. Or so it seems to me.
If I am right, “government censorship” is a specification of “censorship,” as is “parental censorship,” “school censorship,” etc. “Social media censorship” would simply be censorship by a social media company. If this is wrong, we need another term for what the other agents just named do when they limit speech. That’s fine, of course. It’s a mere conceptual matter, one we needn’t worry too much about—what we are really interested in, I think, is whether social media companies or other private agents should seek to silence anyone. Still, if this is not censorship because only speech limitation by government is censorship, then “government censorship” is redundant—and I do not think it is.
That I think social media companies sometimes engage in what is properly called “censorship” does not mean those companies do anything wrong. Free speech is valuable—and so, I think, the first amendment leaves the US more or less absolutist in forbidding government intervention in speech. But that doesn’t mean private agents can never morally limit speech. Of course they can. Of course we can. For example, I stop my son from using certain words that are not appropriate for polite society. I censor him. There are also certain speech acts I would forbid in my classroom if I had to, but thankfully don’t—they don’t ever seem to come up; that is, my students don’t use them (in the classroom, anyway). Similarly, book burning (in some circumstances) by private individuals and book banning in private schools are likely forms of censorship. They’re both legal, even if disturbing.
Some censorship is not only permissible, but expected and probably morally good—disrespectful speech in the classroom, for example, is something we do well to make unacceptable (through non-legal, social means). Is censorship by Twitter, Facebook, etc, of President Trump and his followers good? I don’t honestly know. I am conflicted. On the one hand, I generally agree that more speech is the way to counter bad speech and that airing all views is likely to leave the bad (morally and epistemically) views with fewer believers. And (on the same hand), I worry that people are too often attracted to beliefs they are told they shouldn’t have (the “taboo effect”). Certainly, letting people discuss racist and anti-Semitic views hasn’t (yet) stopped them from spreading and letting people discuss conspiracy theories about fraudulent elections—for which there is no evidence—hasn’t stopped them from spreading. On the other hand, I don’t have any significant doubt that President Trump lies and that his followers are mistaken about a number of important factors, including the supposed fraudulence of the election, and preventing the spread of those false beliefs seems worthwhile. And, I admit, I simply love that in our society government officials face limits imposed by private entities. Corporate CEOs can tell the President of the United States that he can’t use their service; this is not something one can say in Russia or China.
Conclusion: like it or not (and I am conflicted), Twitter and Facebook do not violate any constitutional rights by censoring the President and his followers. As I said previously, this is a matter of property rights. Twitter and Facebook own their platforms just as I own my home. Just as I can forbid someone from entering my home to tell me why Nazi’s were right—or anything at all that I don’t want to hear—Twitter and Facebook can forbid people from using their platforms to say thinks Twitter and Facebook do not like. Twitter and Facebook have the right to censor those using their platforms. Whether they should or not, I cannot presently say.