I've been to Arizona and passed the checkpoints run by the border patrol (federal, not state). They just ask one quesiont, "Are you US citizen?" When you answer "Yes" they'll just wave you through.
Now the law allows the local police to arrest you if you don't look like a citizen. This would require anyone who is not white to carry an ID, even if you are a citizen. It's a real discrimination against nonwhites. It's very similar to a law in South Africa during apartheid. It's a way to limit the ability for nonwhites to move around in the state.