You can do the training anywhere, the instructor for the required dual training must be an FAA CFI ( who might be a Euro local who happens to hold FAA instructor certs).
You may be able to take checkrides overseas if you can find an FAA designated examiner. There may be a few on US military bases. I suspect it would be easier to do the checkride in the US, but get a few hours of preparatory flight training in the local airspace where you intend to to take the checkride, with a CFI who is familiar with the examiner in question. Most US schools have relationships with examiners.
For the written tests (which are more practical, and less theoretical in the US) you may also be able to take them overseas but it might only be available on a US military base. I would contact a few local schools to see of they know of any way to take FAA written tests.