As state agents, all public colleges and universities are legally bound to respect the constitutional rights of their students. That the protections of the First Amendment apply on public campuses is well-settled law.
Private universities are not directly bound by the First Amendment, which limits only government action. However, the vast majority of private universities have traditionally viewed themselves—and sold themselves—as bastions of free thought and expression. Accordingly, private colleges and universities should be held to the standard that they themselves establish. If a private college advertises itself as a place where free speech is esteemed and protected—as most of them do—then it should be held to the same standard as a public institution.
Furthermore, private colleges and universities are contractually bound to respect the promises they make to students. Many institutions promise freedom of expression in university promotional materials and student conduct policies, but then deliver selective censorship once the first tuition check is cashed. They may not be bound by the First Amendment, but private institutions are still legally obligated to provide what they promise. Private institutions may not engage in fraud or breach of contract.
It is important to note, however, that if a private college wishes to place a particular set of moral, philosophical, or religious teachings above a commitment to free expression, it has every right to do so. The freedom to associate voluntarily with others around common goals or beliefs is an integral part of a pluralistic and free society. If a private university states clearly and publicly that it values other commitments more highly than freedom of expression, that institution has considerably more leeway in imposing its views on students, who have given their informed consent by choosing to attend.