That is actually a good question. I would imagine the government would like to be informed. I am just not sure where you would report it, maybe the medical board? I know they have a public website anyone can access that is in English as well.
I know there are tons of horror stories with plastic surgery gone wrong online but Korea is like the world hub for plastic surgeries and they have some of the best doctors in the world. I would argue they have the highest percentage of them in one location.