The idea behind free speech is that people are allowed to put forward new ideas ( especially ideas about how to organize society, what is good and moral etc ) so that people can consider and accept or reject them. The idea is that no entity has a monopoly on truth and if you want to propagate your ideas, make your arguments persuasive and refute your opponents' arguments.
Free speech is ultimately a bet on human capacity for reason and goodness. The idea is that good ideas should win adherents and bubble to the top while bad ideas sink to the bottom as they lose followers.
Sometimes, governments are the most powerful enemy of free speech while sometimes other entities can be.
I am not saying free speech is a good thing, but there is no "exclusive to government" limiting principle to free speech.