This banner is hanging near one of my schools. Sorry for the poor quality of my cell phone pictures; it says:
Intensive Control Period of Foreigners' Illegal Extracurricular Work For Forced Repatriation
[원어민 불법 과외 적발시 본국 강제소환 집중단속기간]
And there are telephone numbers for the Yosu Immigration Office, the Suncheon Office of Education, and the Suncheon Association of Hakwon (순천학원연합회). It goes without saying that you should always be careful of whom you let talk you into giving private lessons, because word gets around, and even in my teachers' office my colleagues talk about who's teaching their children on the side. You should be especially wary of strangers in the area approaching you out of the blue, as happens pretty much all the time.