The First Amendment, like other provisions of the Constitution, applies only to government, whether federal or state or local—it does not apply to private actors. So the New York Times cannot violate anyone's First Amendment rights. Nor can Facebook or Twitter.
It's a little bit like when radio came into existence and the government passed legislation to give the government the power to regulate with the Federal Communications Commission, the Fairness Doctrine, and so on. It may be that similar types of regulation, we may now have come to realize, is necessary with respect to social media. The danger, or course, is government intervention is problematic because government can abuse their power. Imagine if Donald Trump had the power to dictate what speech would be permissible or not permissible on Facebook.
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