It turns out that 14 years ago the U.S. Congress passed a law which enables state and local "LEOs" (law enforcement officers) to do exactly that. The only difference is that the federal law says they may and the AZ law says they must.
According to this U.S. News & World Report article, some jurisdictions in the following states already utilize the provisions of this law, known cryptically as "287g," to look for immigration scofflaws: "Florida, Virginia, Oklahoma, Ohio, New Hampshire, California, and even Massachusetts."
So why aren't all our police doing what the federal law says they're entitled to do? A number of police organizations don't want to have anything to do with immigration as they believe, probably correctly, that to do so would make their relations with the Hispanic community very difficult.
You might want to ask your city, county or state to take advantage of federal law 287g when they encounter individuals whose immigration status is doubtful. Hat tip to Lucianne.com for the link to this USN&WR article.