Hello,
My parents (American and Canadian) have been teaching overseas for years, in both Europe and Asia.
My father has a B.Ed in Science Education, an M.Sc. in Planning and an M.Sc. in Information Technology. My mother has a B.A. in Psychology, a teaching certificate, a Diploma as teacher of the deaf and hard of hearing, and a couple of reading certifications.
Which is all to say, you can become an international teacher in Asia through multiple routes. Most of the younger teachers here have their undergrad, teaching certificate, and a related masters. The most important thing, however, is experience (you cannot get a job in China without a minimum of 5 years, as per Chinese law) and some sort of speciality. It is much easier for an IB Mathematics teacher to find a job than for a general elementary school teacher, simply by stint of numbers.
Most of the jobs at major international schools (e.g., Brent in Manilla, ISB in Beijing, SAS in Singapore, SAS in Shanghai etc.) are got at job fairs held throughout the world in January and February (e.g., Search Associates etc.). Typically, schools require at least two years experience teaching in the US (or where ever you are from).
Hope this provides some insight. This is just my families experience, but I think it is pretty typical. Feel free to PM me if you have more questions.