Discussion Forum—A Way with Words, a fun radio show and podcast about language

In your examples, the original question is strongly implied:
Once the (answer to the question of the correct) level of privacy is determined [upon (?)] , the question becomes who needs to know?
(Now that they resolved the question of her dog's food preference by discovering that) Her dog's favorite is caviar; the question becomes Russian or Iranian?

I am sure, however, that you are right that people misuse the phrase. A similar expression that gets misused a great deal -- perhaps to the point of meaning something quite different now -- is "... beg the question ... ." Originally to beg the question meant to avoid answering a question. People now use to beg the the question to mean something similar to " ... the question becomes ..."