It's not a matter of PGCEs or PGDEs, but rather the fact that teaching qualification, much like professional qualifications for doctors, nurses, social workers, lawyers, etc are regional/local rather than international. On another note, I don't think a master's of doctorate anywhere gives you a teaching qualification anywhere. In the UK, the only master's degree that comes with a teaching qualification is the Master of Teaching degree but that is quite different, because that's a PGDE plus a dissertation, and is often not seen as an actual master's degree anyway.
Whether the English teaching qualification, QTS, is recognised anywhere else depends on the country. Within the United Kingdom and European Union, obviously. And as someone has said, some other English-speaking Commonwealth countries as well (and the UK also recognises their teaching qualifications). A country like Singapore does not have a specific teaching qualification and they access each applicant individually as a result (so could recognise a qualification from anywhere or nowhere).
If you are to teach in a private school, an international school, a tutorial school, or a language school, it's really up to the school whether they see you as a qualified teacher. Most likely as long as you're from a first world country they will see you as such, but in reality it does mean that the PGCE is likely not to be all that important to begin with.