I've seen this addressed a lot with "and", but not with "or".
In the three following sentences:
It isn't John's or Mary's fault.
It isn't John or Mary's fault.
It isn't John's or Mary's faults.
Which would be correct, and why?
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I don't know why you think it would be different with or. As always, the clitic 's gets tacked on the end of the entire noun phrase that it is intended to apply to.
So it must be "It isn't (John or Mary)'s fault", with the parens here used only to indicate the scope of the clitic, not to imply optionality.
Flip it around for proof: "It isn't the fault of John or Mary."