"Lawful contact" at first seems like a heck of restriction, but it really isn't. Yeah, a cop couldn't make random traffic stops to check for immigration status because that's not "lawful". But if there were a valid traffic stop, the cop has now lawfully contacted everyone in the car, not just the driver. More importantly, there's nothing unlawful for a police officer to casually walk in a crowd or walk up to someone on the street and ask them how they are doing. Heck, they can even knock on your front door for no reason. It criminal circles, that's simply referred to as an "encounter" which requires no justification and, more importantly, that would be a "lawful contact" under the Arizona law. So everyone is fair game.
The second part of the law requiring attention is the part referring to an officer having "reasonable suspicion" that someone is an illegal alien. That's a term of art used in criminal law which normally allows for an officer to detain, not arrest, someone if the officer "has a reasonable suspicion that criminal activity is afoot." (Legal battles are fought over that every day). But, honestly, the Arizona law really isn't a change in the law at all. If an officer truly had a "reasonable suspicion" that someone is an illegal alien, they could stop and detain them to investigate even if the new Arizona law was not on the books.
But here's the kicker: What exactly is "reasonable suspicion" that someone is an illegal alien? That's the problem. The only answer is that you would have to use stereotypes to justify the detention of any person solely under the new law. But those stereotypes don't work because not only do they apply to illegal aliens, but they equally apply to millions of legal American citizens of Mexican descent. (Hey, I've seen Born in East LA.)
So the bottom line, I think, is that the new law really doesn't change the law at all. However, you've never seen "suspected" illegal aliens detained on the streets in the past solely to check for immigration status because the cops know there's really no way they can articulate why the person they detained couldn't have just as easily been an American citizen. And the fear that it might start now kind of has The Left up in arms. It is American citizens, who might not look like "traditional America", who will suffer.
Edit #1 to respond to comment: Even if you are here illegally, you still have the same rights to be free from an unlawful detention as an American citizen. You may not like it, but that's the law. And as an American citizen not doing anything wrong, I have a HUGE problem with being detained.
Edit #2: Someone pointed out that being an illegal alien leads to deportation which is a federal civil, not criminal, matter. So maybe cops, absent the Arizona law, don't have the right to detain a suspected illegal alien after all. That's a great point.