I think the new teacher at my hagwon must have the record for teaching on a passport without a work visa.
He said he taught full time at a hagwon for two years before going legit and getting an E-2 visa with my institute. I thought it was risky, but he made it sound so routine. He even used a local bank account for monthly deposits.
He's British, so maybe it's easier for them to stay in country long term and work under the table.
Note: I would still recommend against doing this but I was blown away by how easy he made it sound. Maybe the tight economy has created a market for teachers willing to work full-time at hagwons willing to take them on so they can avoid paying full benefits (like housing, pension and severance).