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.
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.