The qualifications to teach English in a public school in China, Korea, Taiwan and Japan are often similar to teaching in a private school in Asia. However, there are some differences and each country has different requirements. In many cases you will be teaching with a co-teacher.
To teach 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.
The qualifications to teach in a public school in Korea are generally the same as in a private school (hagwon). 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.
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 (ALTs) 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.
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.