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So, starting a sentence with ‘So …’ just might make sense sometimes.
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 …”