Your second sentence is true: Private censorship isn't the same as public censorship; you have no fundamental right to free speech in a context owned by someone else (private website, physical business, physical home, etc.).
But your first sentence needs adjustment: Free speech is something a government takes away from its citizens. It's a fundamental right held by everyone in the world. There are governments that fail to protect it, and of course many that actively violate it.
It irrationally aggravates me when people say this, but: This. Rights are inherent to people, not something granted to them by the whim of people with power.
You are correct. However, these days significant portions of our communications are carried, curated, and distributed only through corporations - to the point where many people rightly feel those corporations have at least as much effect on our life as the government, without being held to the same standards of course. Expecting only governments to honor rights such as free speech does feel a little anachronistic in a world that's increasingly controlled by powerful economic interests. Restricting the obligation to honor human rights to governments is starting to look like a giant loophole, a legislative bug emerging from a legacy system.
Though I do not share these concerns as far as message board moderation is concerned, I believe there is often a good reason why people tend to invoke free speech instinctively in these cases. The expectation is that basic rights are not so much special allowances patronizingly handed down from the government intended for an increasingly narrow context but should instead be understood as innate personhood rights enabling people to live a free and dignified life.
But your first sentence needs adjustment: Free speech is something a government takes away from its citizens. It's a fundamental right held by everyone in the world. There are governments that fail to protect it, and of course many that actively violate it.