Foreign English teachers must satisfy 3 masters. The school, the parent and the government Foreign Expert Office. Remember the FEO can only approve someone where there is no native Chinese that can do the job, regardless of the position. (Ergo Foreign Expert) This applies to any foreigner in any job.
Teaching without a F.E. Cert. will get you expelled, and possibly fined. Also they are to most critical of hiring a non-white, non-native speaking teacher. The school has to push hard to get work permits.
In private schools parents also feel for the money they are charged they want white, native speakers. It is pure racism. It is not by accident that the teacher’s desks are in a glass walled office by the front door. They are referred to a ‘Dancing Pandas”.
They will hire part time foreign students, but will hang them out to dry if gov’t. officials show up (which happens). They pay for their own protection, but not yours.
While they seem to pay better, they require a 40 hour week, week ends and nights (when student are not in their regular school) and offer fewer paid holidays. I taught 1 Sat. class (at 10 A.M) and 2 night classes in 9 years. The most I ever taught at a uni was 14 hours a week at about 1/2 -2/3 the salary. I had summers free to vacation. (Have you been to Mt. Everest?, Angkor Wat?)
R/T airfare was included. I don’t know what private schools do about that.
In short, I don’t recommend you try to teach under the table. That requires a Foreign Expert Certification, period. Having said that I did often teach private students under the table.
I highly recommend the experience. Just play by the rules.