Can Xの担当 have both of these meanings?
If so, then without context to disambiguate the two meanings, is one of the two more commonly used?
For example, how would you translate the sentence Xの担当を決めたら？To me, it could be either "How about you decide who is in charge of X?" or "How about you decide what X is in charge of?" If both are grammatically valid translations, then is one of these more commonly used than the other, or do you have to completely rely on context to decide which one?
Also, this is a vague followup, but does the answer to this question also generally apply to the construction (noun)の(verb)? Does NのV mean "something that N does V to" (N is subject) and "something that does V to N" (N is object)?