In 2010, Barack Obama signed "FATCA" (which has since been widely derided as a disaster) into law. FATCA essentially forces all banks in the world to help the US identify Americans who have money outside the US. It works to some degree, but it also is easy to circumvent (i.e. not using US identification when opening the account, etc)
Nonetheless FATCA still provides some level of information about Americans whose finances are outside the country.
So my question is, how was this done historically - not only before FATCA per se, but before computers, rapid communication, and technology ? How did the US government in the 1950s ensure Americans living in Australia were properly reporting their income and paying their taxes?