Who is obliged to believe what and on what grounds? (Part 1)

Blogger Drulogion has posed quite an interesting question about the relation of belief and the authority of the church. Dru starts with a story:

I once heard a story about a 16th century Jesuit mission in Brazil. After a long interaction with the people and some individual conversions, the chief decided to become a Christian (presumably bringing with him the rest of the tribe). As part of the process, the chief met with the leader of the mission to be examined. The priest began asking doctrinal questions. He asked the chief, “How many natures does our Lord have?”

The chief responded, “As many natures as you say he has, Father.”