This important point is explained clearly here:
We do not derive our right to freedom of speech from the Constitution. More specifically, it does not “come from” the First Amendment.
[. . .]
The Constitution is not the source of our right to freedom of speech because freedom of speech is an unalienable right. What the First Amendment can do is recognize that already existing unalienable right by forbidding the government from abridging it. And that is precisely what it does.
We could put it as follows. The First Amendment does not confer, but protects, the right to free speech, a natural right that is logically antecedent to anything conventional such as a human document or the collective decision of a legislative body. It protects this right against government infringement.
There are two points here that ought to be separately noted. One is that the right is protected, not conferred, by the Constitution. The other is that the right is protected against government infringement. The government may not infringe your right to free speech, but that is not to say that I may not.
Suppose you leave an offensive comment on my weblog. I delete your comment and block you from my site. You protest and cite the First Amendment. I point out that said amendment protects your speech against government infringement only, and that I am no part of the government.
If you insist that you nevertheless have a right to express yourself, I will agree, but add that your right to free expression does not entail any obligation on my part to give you a forum.
Finally, to speak of the right to free speech as a 'constitutional right' or as a 'First Amendment right' can be misleading inasmuch as someone might be led by these words to suppose that the right derives from the Constitution. So it is best to speak of it as an unalienable or natural right.