Is it possible to get the distribution of users by country for a site?
In general, no, at least no easily obtained accurate estimate. You may be able to guess, but it needs a lot of work (or some smart data mining tool going through data dumps, but then making that tool is probably a lot of work too).
What you could, alternatively, at least in the case of Politics.SE, is to to go through the top users (which can define in a number of ways using SEDE, e.g. by reputation, by most questions asked, by most votes, etc.) and then look for clues.
Many users on Politics.SE have dropped clues, which can be found in their profiles (either in the location field, or in their profile description) or in their posts. For example, users asking about and evidently familiar with Sweden are most likely Swedish and probably not American (which you aim to find out).
First, you sample a list of users and check for clues manually. Then, you should have three counts: those who appear to be from the US, those who are probably not and those who left no clues. By comparing these three counts you might be able to make a reasonable estimate.
Going by which questions users answer is not a good metric, many European users answer questions about the States and vice versa.