There's been a lot of talk about social media companies like Facebook, and their occasional removal of user content. This removal is unilateral and not subject to any sort of judicial review, so it's often called censorship.
Is this illegal? Do the social media companies have a duty under the First Amendment to not censor users?
Is this illegal?
No, subject to some possible narrow exceptions discussed below.
Do the social media companies have a duty under the First Amendment to not censor users?
No.
Indeed, usually, there is greater liability exposure for failing to censor content, for example, by failing to honor a "take down notice" under Title II of the Digital Millennium Copyright Act based upon an alleged copyright violation, or for failure to censor content related to potential sex trafficking.
The First Amendment to the United State Constitution (which applies to state and local government via the 14th Amendment to the United States Constitution) is a limitation on the power of governmental actors only.
This said, some state constitutions, such as California's, provide free speech protections not just from government action, but also in spaces that are privately owned, but are open to the public and constitute de facto public forums. The authority of California to expand its state constitutional protections to these private settings was confirmed by the U.S. Supreme Court in the case Pruneyard Shopping Center v. Robins, 447 U.S. 74 (1980). It is conceivable that these doctrines could be expanded to public Internet forums in the case of California based social media companies (e.g., Facebook has its headquarters in Menlo Park, California; so does its sister platform Instagram; LinkedIn is based in Sunnyvale, California; and Google is based in Mountain View, California).
There are also laws that limit how employers can regulate employee speech in a labor relations context, although most of them don't have constitutional dimensions. It is conceivable that these doctrines could limit social media platform's authority to limit some kinds of speech by their own employees, or in situations where the social media platform looked like it was acting as a mere agent of some other employer controlled by that employer for all practical purposes.
There has also been litigation related to free speech on social media regarding the rights of governmental account holders to exercise the same kinds of account management that is available to other users, implicating the First Amendment right to petition the government. The social media platform operator is not itself the primary target in these cases, but if it simply implements its terms of service neutrally with respect to all account holders, it could be facilitating a constitutional violation by its governmental account holders and could conceivably be held liable for aiding and abetting that violation of the law by a governmental account holder (in the context of a lawsuit for money damages this is a special subtype of something called a civil conspiracy).
The Free Speech clause in the First Amendment limits the government's right to act against speech. But it's not the only clause there.
A private website is not a town square. The relevant thing is freedom of the press.
So in fact, the First Amendment says a great deal about companies limiting user speech. Companies are legal persons and have the right, in fact the obligation, as I'll discuss.
And the press belongs to Facebook, Twitter, Reddit et.al and they can put anything they want on it, with or without you.
Of course, that's not what the users want to hear!
Freedom of the press used to be a big deal when a press cost several years' wage. But last I looked, GoDaddy sells presses for $12.99/year. So there's actually a market solution to the free-speech problem you are discussing, and that's for someone to form their own social media site, and use their genuine free press to delegate to users their own freedom of speech. If the market values that, then this site will win.
Courts have historically given platforms wide latitude to manage UGC (User Generated Content) consistent with the health of their platform. As someone who managed content for a large social media company, it's our job to put the platform first, so we achieve our business goals (which tend to be asymptotic to providing a great user experience overall, i.e. on average for the public at large, not Joe Nut's throbbing ego.)
Anyone else running a UGC social media platform will inevitably have the same problem: user activities will conflict with the health of the platform and you'll have to deal with it. When your business runs on UGC, it's not an option to manage it; it's mandatory.
Ultimately, the answer is for users to own their own presses.