I don’t work illegally. You only need qualifications for work permit sponsorship. If you have an open work permit you can work everywhere except full time at public schools.
I currently work full time at a cram school and part time at a public school. I’ve worked part time at a few public schools, these are the best jobs but are inconsistent if you don’t have credentials for full-time employment.
Full-time teachers at privates schools are supposed to have teaching credentials but no one checks if you have an open work permit. I never took these offers though. Either cos it was a low ball offer or I didn’t like the person interviewing me. I’ve subbed at a few though.
I’ve always had more job offers than time. I’m pretty proactive but also there are probably less than 5 native English speakers with open work rights in Miaoli City so I’ve always had my pick.