How do we use the possessive case (i.e., 's) with "or"?

Solution 1:

The Punctuation Guide, which draws its rules from APA and the Chicago Manual of Style, distinguishes between joint or shared possession and individual possession. In joint possession, only one 's appears at the end, since both of them possess the same thing together. In individual possession, 's is used before each noun phrase, to indicate that each person separately possess at least one of the things.

We were at Stanley and Scarlett’s house. (shared possession: Stanley's house and Scarlett's house is the same house)

France’s and Italy’s domestic policies are diverging. (individual possession: France's policies are not the same as Italy's policies.)

Both these examples (and the others) use and. However, the same logic should apply to other conjunctions. Attend to this usage in the explanatory text on the same page:

You should, of course, observe your publisher’s or instructor’s requirements.

In this case, the publisher may have different requirements than the instructor's. They are not the same. This is an example of individual possession, just like what showed up before. The Chicago Manual of Style, 7.23: Joint versus separate possession also includes an example of discrete possession with or:

Gilbert’s or Sullivan’s mustache

In general, it is hard to imagine an example of shared possession involving or. Outside of logic, or is almost always disjunctive and exclusive (see, e.g., Legal Beagle for an explanation). That would suggest the resulting possessive must be individual.

That said, not everyone follows Chicago, APA, or a website called The Punctuation Guide. In the wild, usage involving individual possession nonetheless may only put the noun phrase after the conjunction in the possessive case (Jason Nice, Sacred History and National Identity, 2009):

Recent scholarship has focused upon France or England's ethnic and/or political origins ...

France and England have different ethnic and political origins, right? Nonetheless, the author and the editor(s) permitted "France or England's." So while the logic in Chicago and others may be sound, outside the confines of a style guide, actual published usage varies. Usage books may bemoan using only one 's for an individual possessive (for example, Bryan Garner spends three paragraphs pointing out errors in possessives involving and in Garner's Modern English Usage, 2009, p. 713), but whenever a usage guide can complain about a phenomenon for three paragraphs, that means the practice is widespread. Make your own decision on whether that is appropriate.

Solution 2:

Whether to say:

Is it John or Jane's car? [1A]

or:

Is it John's or Jane's car? [1B]

is the same question as:

Is it the car of John or Jane? [2A]

vs:

Is it the car of John or of Jane? [2B]

The possible ambiguity in [A] sentences is usually resolved in speech by pausing (almost 'John, or Jane') but inserting a comma in writing leaves a different ambiguity - as if we're confused whether a thing is John himself, or Jane's car.

It is correct and resolves any ambiguity to repeat the preposition per [2B], which contracts to repeated apostrophes in [1B].

[B] is apparently referred to as 'parallel structure', and avoids the issue of 'syllepsis' in [A] sentences.