If they're residing in the other country, they're supposed to be paying income taxes to that country, in addition to taxes to the US. The taxes to the US will probably be canceled out by the tax exemption they can claim for the taxes paid locally though, but they still have to file in the US unless they renounce their US citizenship.
Very few people pay income taxes to a foreign government that they reside in unless they are actively working and collecting a paycheck. A passive income stream such as collecting social security or a pension is very rarely taxed. Even things like collecting rent as a landlord back in the home country or earning interest on a savings account there, are also very rarely taxed.
Then they're cheating on their taxes, and that government should probably look into how that person can afford to stay there with zero income. A lot of countries have tax treaties with the US to cover this sort of thing.
