My take on this is that, since this is a quasi-medieval society, all of the organizations might be expected and have a duty to uphold the law wherever they are. They are all, in a sense, military groups. The only differences are where their jurisdictions lie and their reach.
Stone Torches are the most limited, since their authority starts and ends with mining operations. Brass Blades follow by being the police of Thanalan and Ul'dah (most of the NPC guards in the city are Brass Blades and not Immortal Flames). The Sultarnsworn are basically a sort of praetorian guard who answers to the Sultana and so they can probably just barge in and trample all over the other two groups.
But they are also charged with keeping the law in the upper, richer levels of Ul'dah, as evidenced by all NPC guards in the Husting Strips being Sultansworn. The Immortal Flames' are basically the military, being in the outskirts of the Ul'dahn nation and defending it from foreign threats.
With all that said, I think that Sultansworn and Immortal Flames who witness crimes in, for example, an Ul'dahn alley will have the authority and even the duty to take immediate action. However, what they don't have is the authority to take matters out of the Brass Blades hands (unless it's a matter of military or royal importance, naturally). This means that they can make an arrest but their involvement in the matter ends when they hand the prisoner to the proper law enforcement organization. Thing that basically happens when the criminals are thrown in a cell.
Stone Torches are the most limited, since their authority starts and ends with mining operations. Brass Blades follow by being the police of Thanalan and Ul'dah (most of the NPC guards in the city are Brass Blades and not Immortal Flames). The Sultarnsworn are basically a sort of praetorian guard who answers to the Sultana and so they can probably just barge in and trample all over the other two groups.
But they are also charged with keeping the law in the upper, richer levels of Ul'dah, as evidenced by all NPC guards in the Husting Strips being Sultansworn. The Immortal Flames' are basically the military, being in the outskirts of the Ul'dahn nation and defending it from foreign threats.
With all that said, I think that Sultansworn and Immortal Flames who witness crimes in, for example, an Ul'dahn alley will have the authority and even the duty to take immediate action. However, what they don't have is the authority to take matters out of the Brass Blades hands (unless it's a matter of military or royal importance, naturally). This means that they can make an arrest but their involvement in the matter ends when they hand the prisoner to the proper law enforcement organization. Thing that basically happens when the criminals are thrown in a cell.