What type of govermment did the Sioux govern with?

Sioux, like most plains tribes were among the most truly democratic of societies.

The term Chief does not mean statesman, nor king, nor even mayor... A chief was nothing more than an influential member of the tribe. And every band had several chiefs if not a dozen or more.

Some Chiefs were influential because of their success in battle, but would not even be listened to in matters of forage or hunting... and vice versa.

Any member of the tribe was entirely free to come and go as they pleased... could join forces with a war leader, or not...

Ideally...

However, power was mediated thru wealth. In Plains tribes wealth was in the form of ponies.

A man who lead successful horse raids would find lots of volunteers, but most of the horses stolen would end up belonging to the man who formed the raid.

An average Sioux family needed between 4 and 6 horses, minimum, just to move camp. Most did not own that many.

They could go to a chief and ask if they could borrow some horses from the Chief, who often had hundreds... and the Chief would always say yes, take as many as you need...

But in exchange the Chief would expect you to back his agendas in council, ride with him on the war path, or give him a lion share of any horses you took while riding his ponies.

In this way Chiefs bartered with tribal members for influence and alliances. And tribal members considered carefully which Cheif they would seek help from so as not to end up obliged to a Cheif they thought was not doing well.