This is a difficult issue to address, considering the historic plight of Native Americans and the highly controversial nature of the relationship between the United States government and native tribes. Native Americans were not officially granted US citizenship until 1924, and some states continued not to recognize their legal rights as full citizens until the late 1940s.
The United States government has designated almost 600 locations in the country as sovereign Indian territories, which means that Native American leaders in those areas can establish their own laws, public services, taxes, and other rights left to the other 50 states. What they cannot do is establish their own military or issue their own currency, which puts Indian reservations in the same subservient position as states and overseas territories. In that sense, the people who live there are considered US citizens who happen to live on property that the government technically holds in trust.
It may be more accurate to think of Native Americans as US citizens who must pay federal taxes, obtain US birth certificates, register for a US Social Security number, and carry a US driver’s license for identification. This is considered to be a separate issue from nationality per se, which means that someone living on tribal territory is always free to consider himself or herself a member of a recognized tribal nation.
In the strictest legal sense, Native Americans do not have dual citizenship between their tribal lands and the United States, since the tribal territories are not recognized as separate and sovereign nations. They are considered “domestic dependent nations” by the US government. A controversial Bureau of Indian Affairs still addresses issues between the federal government and individual tribal leaders, not the office of Secretary of State. This lack of official recognition as independent sovereign nations continues to be a source of contention between some native political leaders and the US government.
If an individual tribal territory should succeed in obtaining true sovereign nation status, the people who live there may indeed be in a position to claim dual citizenship, with duplicate legal documentation and equal rights to participate in political elections and other civic duties. The question of loyalty to a particular nation may become an issue if this new nation chose to raise up its own military within the geographical boundaries of the United States, however.
Native Americans are generally considered to be citizens of the United States or Canada first, then legal residents of their chosen tribal lands second. It would be the equivalent of a state resident being legally considered a US citizen but residing in Ohio or Nebraska or California. Those states are allowed to form their own governments and enforce their own laws, but they are still subject to the rules and regulations of the United States as a whole.