In this question the purpose of the Jedi is discussed but no one mentions their true function in society or the way that their order is funded, since they have access to supplies, technology and have temples all around the galaxy, not to mention that they are associated to the Government through the Jedi Council, but are not supposed to interfere in political decisions or to have as much power as the Senate.
I found an interesting analysis that describes the Jedi as an autonomous religious cult to which the Republic outsources its police and military power, much like a mercenary army in our world (or perhaps, like the Sōhei in ancient Japan?). Being mercenaries explains part of their funding and power, but although we see them leading the Clone Army, that is indeed an extraordinary case, as Mace Windu states in Episode II (0:26): We are keepers of the peace not soldiers, similar to Obi Wan's "keepers of peace and justice in the Old Republic" definition. This sounds very much like police, but apart from Obi Wan making detective work to find Jango Fett in Episode II we never see a single Jedi making something analog to what a Police Officer does.
So what are the functions and responsibilities of the Jedi in relation to the life of the average citizen during the 1000 years before Episode I, since there were no Sith Lords (or rival religious groups) to hunt in those times? Did they make religious services and ceremonies like our priests and monks and/or acted like a judiciary branch of the government or were just ordinary law enforcers, much like the Stormtroopers during the Empire?
The wookiepedia article that says that they pursued wanted and dangerous criminals, pirates, and smugglers goes with the just ordinary law enforcers part and the Jedi organization and population presented in the first three episodes seems too big for that petty job. What I'm asking about is more about their role in the everyday world, as a citizen of Coruscant or any other planet would see them, if they find them on their way.
I expect to find a good answer from the Legends continuity, but I prefer if the answer comes from the official Canon.