There is no difference about working for a Government school or a private school. So long as the school operates under the Government agency OBEC then the rules are the same. This covers about 95% of schools, both private and Government.
To obtain a work permit you will need a teacher licence or a "waiver". Both are obtained from the Teacher's Council of Thailand(Khurusapa)
Without this document you will not get a work permit, and if you were on a Non B visa you would not get an extension.
You need a Bachelor Degree to get this document.
Khao San road does not work anymore. There is one guy that I know of that got a 3 month, suspended, jail term. The jail term was suspended because he was only found out FIVE years later when the Government did a sweep of previous applications. He was arrested when he was extending his retirement visa, and I repeat FIVE years later.
The "not valid" degrees that have been mentioned are "life experience degrees". Some work some don't and it all depends on who deals with your application at the Teachers Council.
Like Nookiebear said, and he is quite correct, if they decide to delve then you could be in trouble.
You may like to try a Language Centre, there are a couple in Buriram, or a University. These establishments are not covered by the Teachers Council so you do not need this extra document.