You cannot legally teach private lessons in South Korea. But once you get here, you might be directly off the plane when someone first approaches you: “Can you teach my son/daughter/niece/nephew/friend?”
It sounds appealing, and it is. You can make A LOT of money teaching private lessons (payment starts at W50,000 an hour). But, it’s important to understand that you can also face fines and even deportation (in the most extreme cases).
It is important to know the legalities of private lessons in Korea before you start signing up for that private lesson every other day after school. When you teach in South Korea, the company or school that sponsors you for your visa is completely responsible for you for the duration of your contract. The visa you get is specific to that school or company and legally you cannot work anywhere else.
With an E-2 visa in Korea (a teaching visa) you can only legally work at the school that sponsored you. If you want to work elsewhere legally, you must obtain a “rider” or addendum to your visa- which entails getting permission from your full time employer and registering your part time job with your regional immigration office.
You can also try and obtain permission from your school to teach lessons. But, in all likelihood, they will say no.
In my experience, it has been a “don’t ask, don’t tell” policy. Many friends have been set up with private lessons through their employers. And maybe this is a safer option. But just remember, it comes with risks.