Because of the 14th Amendment, the states are bound by the 1st Amendment's prohibitions against abridging the freedom of speech.

"Freedom of speech" is a concept that is bigger and more far reaching than the government. If someone threatens to punch you in the face for saying something that he doesn't want you to say, he has violated your freedom of speech.

If your employer fires you for something that you said outside of work and separate from your duties as an employee, that employer has also violated your freedom of speech. The notable difference being that in most circumstances, there's nothing illegal about that.

The point that I'm making here is that it's absolutely possible to suppress suppress freedom of speech without violating the law.