How were Native Americans politically organized before the arrival of the Europeans?
If I'm not mistaken the chief was head of the tribe and the shaman also held some power, but what about the other tribe members?
The previous post was very accurate. It's hard to politically classify Native Americans because there were so many tribes and expressions of the political good. I think that one example of this might have been the League of the Iroquois, where multiple tribes unified their divergent interests and unified as a political entity. Such a political reality brought the idea that heterogenous articulations can find common ground under one "roof," in a matter of speaking. I think that this political organization reflected that Native Americans' grasp of political realities greatly influenced the framers and their ideas of how multiple groups or interests can find a sense of common ground without difference being automatically translated into violence or intense antagonism.
Keep in mind there were hundreds of tribes scattered across the Americas, and that the exact form of government they practiced varied from place to place and people to people. But many tribes practiced a council form of government. The Council consisted of the tribal elders, the Chief, and the Shaman, along with some warriors. There were War Councils (like today's subcommittees in Congress, but more efficient) that dealt with emergency threats.
The Chief's word was law, and the Council served as advisers, although their advice usually carried weight.
The Judicial Branch of the tribes was the whole tribe. The Chief or the head of each household would mete out punishments, and the tribe lived according to an old and clear code where everyone knew what was expected, and what punishments would be for offenses. These punishments were enforced by all tribal members.
Since native peoples were scattered, they were loosely organized into bands, tribes and nations that could cover tens of thousands of square miles. Some bands of the same tribe might never come into contact with one another, or at least very rarely. In situations where the threat was to the entire nation as opposed to just one band, sometimes councils of the chiefs of the different tribes would take place.