Starfleet personnel ends up back in time on a Starfleet vessel. We both serve the same organization. My duty is to protect the timeline I come from. Your duty seems, implicitly, to aid a fellow Starfleet officer in their mission (to protect the aforementioned timeline).
It seems like Starfleet should have a dedicated Temporal Security crew on every starship and starbase for such an occasion. You find a supposed time traveler, you immediately call this team. They sequester the intruder and go through a careful interview to verify their claim as cleanly as possible, then render what aid is needed to secure the timeline and get them home (or, barring that possibility, get them somewhere isolated where they can’t contaminate the timeline). Then, maybe memory wipe the Temporal Security team (and possibly anyone else who interacted with the traveler). On the flipside, if you end up back in time, it’s expected you should immediately attempt to contact the local Temporal Security crew.