Several years ago in this area, the Snohomish County Sheriff issued deputy commissions to the members of our local Tulalip Tribes police department, making it clear that the Tribal police had jurisdiction over everyone on the reservation, tribal members or not. So Tribal police can arrest anyone on the reservation. There are no more free passes just because you happen to cross the tribal boundary. Before that, if you were not a tribal member and committed a crime on the reservation, the Tribal police could detain you for a short period of time but had to call the Sheriff or state patrol to actually arrest you.
After reading this thread, I wonder if this approach is uncommon.