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As SE's have neither username nor password, they'll never encounter content B.
Here's a slightly tangential answer: As a human user, it annoys me to no end when I find something in a search engine, go to the referenced site ... and run slap into a paywall or login prompt. If content isn't accessible to random users off the street, don't let search engines know it exists.
On a page we find content A - if a user logs in, the content changes slightly to content B. Need I worry about duplicate content?