If someone was caught stealing from a band of Native Americans out west, say - what would happen? Could/would the Native Americans exact their own justice, or did they have to turn the thief over to the American government? What kind of treatment or punishment could someone accused of thievery expect?
I can answer a very small part of this, regarding the Navajo. I do want to stress that the Navajo, by 1900, were by no means a "band". And instead were generally dispersed agriculturalists and ranchers, like many whites in the region, and had a strong system of leadership and government.
As per the 1868 Treaty of Bosque Redono that established the Navajo Reservation, Navajo who committed crimes would be tried and punished under US Federal law and not tribal ones:
"If bad men among the Indians shall commit a wrong or depredation upon the person or property of any one, white, black, or Indian, subject to the authority of the United States and at peace therewith, the Navajo tribe agree that they will, on proof made to their agent, and on notice by him, deliver up the wrongdoer to the United States, to be tried and punished according to its laws."
Now what this meant was complicated' and the 1900s is a long time period. At the time of the treaty, wrong-doers were to be turned over to the US, which meant either Bureau of Indian Affairs or US Military officials. However, as the reservation expanded and the Navajo became more of a sovereign nation, they were allowed to keep their own courts that abided by US Federal Law.
By the time of the Navajo Livestock Reduction in the 1930s, most court cases were held on the reservation. And when men were not convicted by Navajo juries for refusing the reduction, they were extradited with strong protest by the Navajo community to courts in Arizona and Utah that were more than happy to convict them. This is part of a trend of racism caused many issues with the Navajo and nearby justice systems. Skin color certainly played a large role in punishments.
I do not have specific examples of non-Navajo and the Navajo justice system on hand and unfortunately do not have time to comb archives right now. However, the Navajo Nation Police Force was founded in 1959 and modeled on other US police at the time. Navajo and non-Navajo on the reservation were and are subjected to their law enforcement.
Again, I'm sorry I can only answer part of this but I hope it helped.
Sources: The Treaty of Bosque Redondo
The Roots of Dependency: Subsistance, Environment, and Social Change among the Choctaws, Pawnees, and Navajos by Richard White
Navajo Livestock Reduction: A National Disgrace by Ruth Rossel and Broderick Johnson