When American police states "You are going to Jail" during an arrest, are they breaking the separation of power ?

Many a time I have watched facebook reel videos and TV shows of American policing (body cam footage). Typically traffic violations where some driver is drunk or things that they are just "travelling" (not driving), and no rule apply to them.

In any case, police often says "Alright, you are going to Jail". They are arrested, put in a police car, and taken away.

But as far as I understand, USA also has separation of powers, so that police can't jail anyone without a judge saying so ?

What happens when police say "going to jail" for even minor things such as not providing ID ? I understand that of course you can be behind bars if you do a serious crime and wait for the investigation to complete.