When Qui-Gon Jinn and Obi-Wan Kenobi were sent on a mission to deal with the Trade Federation on Naboo, they were technically operating as agents of the Republic. But after a topsy-turvy journey brought them to Tatooine, the Master and Padawan returned to Coruscant and reported directly to the Jedi High Council. So who do the Jedi answer to? Themselves? Or the Republic? The answer seems to be both, but within that slippery pluralism, the most influential Jedi of all may not wield the most practical power.
