Native American land is controlled by tribal law. Tribal Law only applies to those who live on the Reservation. However, if you are carrying a concealed firearm with a valid permit in the state where the reservation is located, that permit may not be valid on the Reservation. If caught, they will most likely take the firearm and tell you that you can go to Tribal Court to get it back. Some Tribes consider federal and state highways through their land as being under their control and say you cannot carry a firearm in a vehicle or on the roads on their Reservation. Tribal Police, in most instances, work closely with the Local Law Enforcement surrounding the Reservation. If you break a state firearms law, they will most likely detain you and contact local authorities.
We recommend that, before you carry a firearm on any reservation, you contact the tribal leader and, if possible, obtain written permission. Otherwise, keep it unloaded and secured in your trunk or locked box in the back of a vehicle that does not have a trunk.

At present, there are 574 federally recognized American Indian and Alaska Native tribes and villages.
Approximately 56.2 million acres are held in trust by the United States for various Indian tribes and individuals. There are approximately 326 Indian land areas in the U.S. administered as federal Indian reservations (i.e., reservations, pueblos, rancherias, missions, villages, communities, etc.). The largest is the 16-million-acre Navajo Nation Reservation, located in Arizona, New Mexico, and Utah. The smallest is a 1.32-acre parcel in California where the Pit River Tribe’s cemetery is located. Many of the smaller reservations are less than 1,000 acres.
Because the Constitution vested the Legislative Branch with plenary power over Indian Affairs, states have no authority over tribal governments unless expressly authorized by Congress. While federally recognized tribes are generally not subordinate to states, they can also have a government-to-government relationship with these other sovereigns.
Furthermore, federally recognized tribes possess both the right and the authority to regulate activities on their lands independently from state government control. They can enact and enforce stricter or more lenient laws and regulations than those of the surrounding or neighboring state(s) wherein they are located. Yet tribes frequently collaborate with states through compacts or other agreements on matters of mutual concern, such as environmental protection and law enforcement.
Source: Bureau of Indian Affairs FAQs
https://www.bia.gov/frequently-asked-questions

Utah Division of Indian Affairs
Utah is home to eight distinct tribal nations, each with a unique heritage that can be found among the state’s many sacred places and extends across Colorado, Arizona, and Nevada.

Note: A U.S. Supreme Court Ruling in United States v. Cooley (May 2021) held that a tribal police officer has the authority to temporarily detain and search non-Indian persons traveling on public rights-of-way running through a reservation for potential violations of state or federal law.

Tribal Law Enforcement
Cross-deputization agreements
Allow law enforcement personnel from state and tribal entities to cross jurisdictions in criminal cases. Cross-deputization agreements have been used to enhance law enforcement capabilities in areas where state and tribal lands were contiguous and intermingled. Under some agreements, federal, state, county/local, and/or tribal law enforcement officers have the power to arrest Indian and non-Indian wrongdoers wherever the violation of law occurs.
Tribal police powers
Authority to exercise criminal jurisdiction over all tribal members and the authority to arrest and detain non-Indians for delivery to state or federal authorities for prosecution. These tribal police powers are generally limited to tribal lands.

Additional Information