The qualifications to teach English in a public school in China, Korea, Taiwan and Japan are often similar to teaching in a private language school or academy in Asia. However, there are some differences and each country has different requirements. In many cases you will be teaching with a co-teacher.
Teaching English in a public school in Taiwan
The requirements to teach English in a public school in Taiwan you need to be a certified teacher in your home country. That is not to be confused with a TEFL/TESOL certificate.
Teaching English in a public school in Korea
The qualifications to teach in a public school in Korea are generally the same as in a private school (hagwon) except as of recent you need to get college transcripts and if you don't have a teaching license then you may need a language teaching qualification.
Programs such as EPIK, SMOE and GEPIK place teachers in schools across the country. You can also get hired directly through a public school without going through any program. These jobs are often listed by recruiters.
You may also enjoy reading this post about public school and hagwon differences.
Teaching English in a public school in Japan
The qualifications to teach in a public school and in a private school (eikaiwa) are usually the same. A popular program provided by the government is the JET program. The JET program places foreign teachers (ALT's) throughout the country in public schools. Interac is a private company in Japan that does something similar.
Read more on teaching as an ALT teacher in Korea and Japan.
Teaching English in China in a public school
The qualifications to teach in a public school in China are generally the same as in a private school. Some private companies in China will place teachers in public schools to teach. In some cases one does not need to have a degree to do so if hired through a private company.