The whole concept of "Human Rights", both the "first generation" rights, like the freedom of speech, and the "second generation" rights like universal access to education, come mostly from the French Revolution (the Declaration of Humans and Citizen Rights, to my knowledge, predates the 1st Amemendement).

If you think legal documents are important, this is backwards. Both were written in August 1789 and probably had little influence on each other. But both were largely based on George Mason's 1776 Virginia Declaration of Rights. Mason went on to write the US list, after refusing to sign the Constitution in 1787 and demanding a national list. In particular, his 1776 document gave freedom of press. I don't know how they expanded from press to speech; maybe one of the 1789 documents copied the other in this expansion, but Milton bundled the two in the 1644 Areopagitica as did many later people. As to the Enlightenment concept of human rights, it is pretty clear in Jefferson's Declaration of Independence. But it owes a lot to pre-Enlightenment English documents, particularly the Petition of Right of 1628.

But why should we trace these concepts to their endorsement by governments? The documents did not create the ideas, but adopted them from a long train of authors, particularly the French Enlightenment. The documents are important as signposts in the triumph of the Enlightenment, but they are very crude measures because talk is cheap.