As a corollary to this government vs. private distinction, a private institution can facilitate and encourage a climate where hate speech is condoned from the top down.
As a faculty member of 11 years with tenure, I was fired from a Mormon University because I defended a professor accused of homosexuality and in appropriate conduct. Complete fabrications condoned by the admin. and therefore ritualistically followed by the faculty of automatons. I was the only one who took a stand against the free speaking of lies, deceit, and prejudice. And, I could do nothing because they were a private organization.