Government-Run Fora on Private Platforms, in the @RealDonaldTrump User Blocking Controversy
Whether the First Amendment applies generally turns on who is imposing a restriction (the government vs. a private party), not on whether the speech is on public or private property.
Some readers have asked how it can be that Twitter can restrict tweets, and ban users for tweets it sufficiently dislikes, but @RealDonaldTrump can't.
That's because Twitter is a private entity, which isn't bound by the First Amendment. (Whether Congress could constitutionally impose some sort of constraints on Twitter's banning users is a separate matter; so far, Congress hasn't.) President Trump is a government official, and his actions are potentially constrained by the First Amendment.
One can argue that he's running the account in his individual capacity; but once the court concludes that he's acting as an official, and not just as a politician, the First Amendment does apply to him. And if the blocking were done on a government entity's account, such as an account run by a school board, a police department, or a city council, then the First Amendment would even more clearly apply.
Nor should it matter that President Trump is using private property for his speech. A city council, for instance, is barred by the First Amendment from kicking speakers out of an open comment period based on their viewpoints. (It has no constitutional obligation to set up such a period, but many city councils choose to.) And that remains so even if the city council decides to meet in a privately owned building that the owner has let it use for the occasion.
Likewise, a public university can't have a policy saying, "we'll issue all students university e-mail accounts, but we'll delete the account of any student who uses the account to say something offensive based on race, religion, sex, or sexual orientation." And that's true even if the university-issued e-mail accounts are hosted on Microsoft or Google computers.
Whether the First Amendment applies to a speech restriction generally depends on who is imposing the restriction. If the government is imposing the restriction, then the First Amendment does apply, whether or not the speech is on private property. Likewise, if a private entity is imposing the restriction, then the First Amendment doesn't apply, whether or not the speech is on government property. (If, for instance, I hit you because of an offensive message that you're wearing on a city sidewalk, I'm guilty of a crime and a tort, but not a First Amendment violation, unless I'm acting in my capacity as an agent for the government. If your private employer fires you because it learns of your having said something offensive on a city sidewalk, then in many states it wouldn't be guilty of anything, and in any event not of a First Amendment violation.)
So, again, one can argue that @RealDonaldTrump is run by President Trump in his personal capacity, not his official capacity; but once the court rejected that view, then it doesn't matter that @RealDonaldTrump is a forum set up on a private company's computers. Though Twitter, as a private actor, isn't bound by the First Amendment, a government official, acting in his official capacity, is bound by it.