In their comment to an answer on the question "Are foreign personal names usually written in katakana rather than Romaji?", user sawa says:
... Chinese names should be written in kanji rather than katakana and read by the Japanese pronounciation. For example, 金大中 is キムデジュン, not きんだいちゅう, but 毛沢東 is もうたくとう, not マオジードン. ...
This got me wondering, since each Japanese character generally has at least two readings is there always one clear reading for pronouncing such names?
I expect of course that the "on" readings would be used, but often there is more than one on reading for the same character.
From the comments so far (no answers yet) this is indeed very interesting, and on Zhen Lin's prompting I would like to include Korean names as well despite their being discounted in the older linked question.