So, no, you're quite wrong on that.
It seems to go both ways. In practice, IIRC, the state governments did pretty much as they pleased, restricted by their own constitutions, until after the War Between the States. Please refer to this page from the Library of Congress.
The First Amendment reads, "Congress shall make no law..." How that language could be construed to restrict state or local governments is beyond me.
The Second Amendment reads "...shall not be infringed." This is much more broad and I'd agree with those who say this amendment already restricts state and local government.
The Tenth Amendment reads, "The powers not delegated to the United States by the Constitution, nor prohibited by it to the states, are reserved to the states respectively, or to the people." This would indicate that the state, and by extension local, governments are less restricted than the federal government. Powers not delegated to (the federal government) are reserved to the states. I do grant that the Constitution can restrict state governments; it reads so right there in the 10th. The 14th seems to make it explicit.
I have no doubt that many states, from the start disobeyed the Bill of Rights. I think this was more do shoddy wording by the authors, not an intention that First Amendment rights didn't cover the States, as well.
So I step back a bit, and say that, in practice, you may be right, but I believe the intentions of the Founders was that the BoR would apply across the board.