It's a double edged sword. If you are not here officially as a teacher then you may be flouting the law. You may have teaching certification but if you are not registered as a foreign expert to teach, then it's not legal. On the other hand, if you are thinking about moonlighting, it may be okay but still illegal if you are here on a alien working permit. As a guide, many teachers provide tutoring on a one on one basis either at the child's home, at a fast food joint or rarely, at their own apartments. Still you need to be careful. All it takes is for one parent to go to the police for whatever reason and you're done, although this is unlikely.

Get your vocabulary right guys please. There is no such thing as a "residence visa". You either have a Residence Permit stating the purpose of your residence (in Chinese), or a Visa, which category indicates the purpose of your stay.

A consulate outside of China grants you a Visa, and a PSB in China grants you a Residence Permit.

The practical differences might be thin in this context, but the difference must still be made.

Now, you must know that, in order to work in China, you must hold a "Z" visa, or a working Residence Permit following a Z visa. Both usually go along with the Working Permits stating your employer. This is the only situation in which working is legal in China.

The legal answer is 'no'.

The usual answer is 'yes'.

Technically, you can't work for a company. But you can do private tutoring and it will not violate the law. Not necessarily legal, but not illegal. But in reality you can teach pt at a school  because they don't care. But understand that you will be breaking the law.