The two issues (English teachers in Taiwan or Japan and teachers of Mandarin/Japanese abroad) are totally different because of differences in the market.
In Taiwan, the idea is to get a white face up in front of your English class. That’s what sells seats and gets parents to sign kids up (and gets adults to sign up for adult classes, too.) But that’s because there are literally thousands and thousands of people who are actively learning (or trying to learn) English in Taiwan. There simply weren’t (still aren’t, but the gap is slowly closing) enough people with the barest of qualifications – being able to speak the language – available to teach all those classes.
In the US (I’ll speak to the country I know), there are relatively fewer Mandarin and Japanese classes. While these programs are growing, they are not growing that fast, and the percentage of all students who take Chinese or Japanese is tiny. You hardly ever find an adult who just decides to learn Chinese to, say, improve his career. So there are far fewer opportunities to teach.
For the informal classes, then, you can still get by with little or no formal qualifications – just being able to speak the language to some extent is probably still good enough. That means one hour here, two hours there, nothing more. I can’t imagine anyone living in a place where there were enough of this kind of classes to make a living from. Pay for informal classes is generally low; it’s usually something people do for the fun of it, or because they hope to eventually painfully develop enough “connections” to be considered for a real full-time tenured job, assuming they have the qualifications to do it.
In the US, some ads for university-level Chinese teaching jobs may ask only for an MA, but that’s the ad. You need to know that those who apply for the jobs almost all have Ph.Ds (although not in Chinese teaching) or are ABD (all but dissertation) waiting to complete their Ph.Ds. I would be very surprised to hear of a person with nothing more than an MA taking a permanent tenure track job at a university. They may hire an MA for a temporary appointment or a “visiting” lecturer spot (maybe) but even for these fairly undesirable (in terms of permanent prospects) jobs, these days they usually get 10 to 20 applications from people with Ph.Ds. As a white girl with a Ph.D in teaching Chinese, I’d prefer that the applicant with an actual degree relevant to the job be considered first, but that doesn’t seem to happen. The jobs go to Chinese people with “a” Ph.D. (usually not in teaching; almost never in teaching, sometimes in linguistics, usually in Comparative Literature) in preference to non-native speakers, regardless of level and credentials.
So, although you don’t face the non-native speaker barrier, without a Ph.D. I can’t be very hopeful for you even if you get an MA in Linguistics, I’m afraid. Lots of opportunities to teach Chinese or Japanese informally (exchanges, night school, extension, adult ed, Boy Scouts, whatever) but relatively few “real” jobs open to those without a Ph.D. (and, I might add, publications and teaching experience – it’s getting more and more difficult to get a job with a Ph.D. and assuming you don’t mind relocating at your own expense.)
It might not fit your personality or your goals, but you could consider doing an MBA or a law degree – either of those, plus native Chinese and Japanese and good English, would put you in an enviable position. Relying only on Chinese to make a living is risky. I’ve managed to do it for 20 years but I consider myself extremely lucky to have been able to do so, and it’s been more through translation and interpreting than teaching.
Sorry for the cold water. Your mileage may vary but at least in the US, I doubt it.