The sense I got from the Prequels is that there was a relationship between the Jedi and the governing body of the Galactic Republic; I mean the Jedi Temple is not only located on the same planet as the Galactic Capitol, but the Temple is right next to the Senate building.
The Jedi don’t participate in the politics of the Republic, but they seem to serve them in things like conflict resolution, security, and intelligence gathering. And when the Clone War started, the Jedi served as military leadership. Though that might be something new, since, as we saw the Republic didn’t have much of a military and needed a newly created clone army to fight its battles.
It’s a bit of an odd relationship; the Jedi are, ostensibly, some kind of monastic order, but they appear to be subservient to the political body of the Republic, and serve in an enforcement capacity. A bit, I suppose, like the Nights Watch from Game of Thrones.