There's no similarity in this debate between accessing something private (that the owner didn't want published) and distributing/sharing new copies (or the same copy) of something published.
Every video copied to Facebook in the discussion here was originally a legitimate video uploaded to YouTube for public viewing.
If you want to continue the actual discussion about the place of the public library versus online sharing of ideas, you'll find only one reasonable distinction: the library isn't for-profit and putting ads on things. Facebook is awful here, but not because videos are being copied and shared — they are awful for their ads and tracking of people and their enabling of plagiarism (which is when someone makes fraudulent claims of authorship and has nothing to do with whether or not copies are made).