Although my forebears are instead from Scotland, England, Germany, I can still imagine a little of what it would be like. I just have to think back to being in China.
The situations are different in one obvious way. In contrast to law-enforcement officers in Arizona, the Chinese authorities didn't have to waste time wondering whether I was a citizen. One glance told them where I stood. (I understand that there are some Caucasian-looking Chinese citizens, but they are scarce.) The only judgment call was whether they should bother to check whether, well, my "papers were in order," in the phrase we all know from WW II movies.
If they had checked very often, I would have been in trouble. In theory, foreigners are always supposed to carry their passports (as Chinese citizens are supposed to carry their identity cards). In practice, I almost never did. When checking in for a flight or registering at a hotel in China, sure: Without a passport, you couldn't do either thing. But when at "home" in Shanghai or Beijing my wife and I kept our passports in our apartment's safe. The theoretical risk of being asked for documents was outweighed by the truly dire potential consequences of our passports getting lost or stolen.
Once, this policy led to minor embarrassment.* Once, it nearly got us into a serious jam.We were in Tiananmen Square on the eve of the 20th anniversary of the uprising there. I took a picture of plainclothes police hustling someone off to custody, which -- perfectly predictably -- switched the police force's attention to us. The initial complaint was "interfering with policemen in their work"; the real problem was not having our passports on us and therefore not being able to prove that we were in China legally (plus flouting a law we should have known of and obeyed).** I have often heard from journalist and activist friends that whenever they got into a delicate situation of any sort, officialdom's first response was to ask for "their papers" -- much as American motorists know that the first thing they'll hear if a policeman pulls them over is a request for license and registration forms.
Here's the point of comparison between the impending Arizona situation and China: it's no fun knowing -- as citizen and foreigner alike know in China, and as Hispanic-looking people in Arizona soon will -- that you can be asked to show proof of your legality at an official's whim. But if it's sobering to think that the closest analogy to a new U.S. legal situation is daily life in Communist China, we should also look on the bright side. With some notable and serious exceptions, I typically did not see Chinese police asking for papers on a whim. Usually something had to happen first. Maybe soon the Chinese State Security apparatus can travel to Arizona and give lectures to local police and sheriffs. They can explain how to avoid going crazy with a new power that so invites abuse. "Civil Liberties: Learning from China" can be the name of the course.
* We were trying to visit a very politically-sensitive exhibit on Tibetan history at the Cultural Palace of the Minorities in Beijing, and on arrival we found that authorities were checking all foreigners' papers -- plus identity cards of Chinese citizens. In the end showing them my Washington DC driver's license, plus our apartment key in Beijing, somehow sufficed.
** I'll omit the details, but eventually they let us go. Those pictures no longer exist, though remarkably I was allowed to keep the camera. The next days, hordes of plainclothesmen finessed the matter by simply opening umbrellas in front of anyone who tried to take a picture or shoot a video in Tiananmen Square. Weird -- in fact, silly -- but effective.