It is really complicated, and I'm not at all an expert, but I believe it is more complex than "they are sovereign, end of story".
Through centuries of conquest and betrayal, the US governments have eroded much of the tribes' previous power, and these days they exist under a complex legal framework in which they have some sovereignty and some autonomy, but not to the same degree that, say, Canada or China have. Their rights under US law are granted by (and can be revoked by) acts of Congress. Lacking their own militaries and international diplomacy, they are more like states than countries as we think of them.
But it's actually way more complicated than that and depends on whether the tribe is federally recognized, what specific rights Congress withheld from them, whether they are prosecuting tribesmembers, other Indians, or non-Indian Americans, etc.
At the end of the day, it unfortunately comes down to who has the bigger stick. Tribes can say no all they want but it's up to the US Congress to listen or not. And they don't necessarily have the same Constitutional protections that US states have, either. It just really depends. And again, I'm not an expert at all!
This page has more detail: https://www.uaf.edu/tribal/academics/112/unit-4/generalprinc...