Every PSB in China knows that companies hire unregistered staff; they only "crack down" when they need money (the Shanghai PSB uses the schools here like ATMs). While it's technically illegal to work on tourist or business visas, plenty of people do; I've known people here in Shanghai who simply run to Hong Kong every three months to update their visa or get their visas through fake shell companies. Likewise, it's extremely easy for any school with the right connections to transfer your visa. I've never had any problems at all with this, nor has anyone else I've known. I've known people who moved from one city to another and simply had their employer transfer their old visa to a new visa; provided you have a letter of release (and, for a sufficient fee, forged letters of release are easy to come by). "THIS IS THE LAW" doesn't mean jack in China; what matters is who you know- or more specifically, who your boss knows. The better the company, the more they can get away with.

Just make sure to get all contracts IN WRITING before you do any work, ask local teachers who's reliable and who's not (in Shanghai, Wall Street, Web International, Market Leader, and Tutor Serve all have decent reputations- I work for several of them, and any public school will be bound by government regulations). The international private schools generally require full teaching certifications, but they offer very good pay and are typically the best jobs for teachers in town. If you're an ESL teacher, though, the real money will be in the tutoring on the side. My salary is 7600 RMB a month, but I can easily pull in another 15000 a month through tutoring. $3500 a month + free apartment + six weeks of vacation a year... not a bad gig.

"And in the big cities, it's just a chaotic, free-for-all party."

You said it. This applies to the job market as well, though. There are no certainties here, and that's all part of the fun of it... but if you can learn to hustle, you can have a heck of a time. Just don't expect this to be like some PeaceCorps project- think of yourself as a free agent, selling your services and navigating the law as you need to. China's here to be surfed.