It's part of the same first amendment that gives you the right to free speech. Go look it up. There is no such thing as privacy in public, and if you feel like you have a right to privacy in public, then you need to read up on the first amendment. The only thing that requires permission is when the footage is used for commercial purposes, then you need permission to use it.
And FWIW, citizens have a right to get the footage the government records. You can get any camera footage from any government building, and even personal cellphones of government emplpoyees if they happen to film something with their personal cellphone while on the job.
The first amendment doesn’t mention recording, although I admit that an argument could be made regarding an implied right. But then why are wiretapping laws allowed? Hiding a recorder in an otherwise public meeting area can violate these laws in some states (notably Illinois and Oregon). Are these laws unconstitutional?
Besides, the first amendment does not give the government the right to record or to benefit from those recordings. Rather, it prevents the government from restricting speech acts of private citizens. Therefore there should be no conflict with the 1st if a hypothetical law prevents the government from recording or accessing private recordings, even if private recordings are considered protected speech.
And FWIW, citizens have a right to get the footage the government records. You can get any camera footage from any government building, and even personal cellphones of government emplpoyees if they happen to film something with their personal cellphone while on the job.