http://volokh.com/2010/05/31/shedding-light-on-the-az-immigration-law/the pdf in the link is a great analysis of the Arizona immigration law.
the upshot: people like the AZ governor and the bill's author who insist the bill does not authorize racial profiling are flat wrong (and, of course, they should know it). At the very least, the bill allows racial profiling, because the bill only forbids racial profiling that would be unconstitutional or against federal law, and not all racial profiling is (for example, in immigration, using a person's race as one of multiple factors in determining whether police have a reasonable suspicion is permissible). At most, the bill may
require racial profiling by allowing private parties to sue the police to force them to enforce the bill. The extent permissible and extent necessary will only become clear after a few rounds of litigation. A complicating factor is that the governor signed an executive order modifying this bill that she said was intended to prevent racial profiling, but the change in the text doesn't do that (but courts, especially "liberal" ones, will find it relevant that she said, on the record, that that was what the change was for, and may decide to honor her stated intent rather than the formal change in the text).
When police are required to stop people and ask for their papers, and when they are not required but may do so, is also not entirely clear. Cops can demand papers from people they have a reasonable suspicion are illegal, but that mostly goes back to what kind of racial profiling goes on.