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

I'm having a hard time finding out what the grammar is when we want to use "or" and possessive. If, for example, I want to refer to the information of a person or an organization, do I say, "that person or organization's information," or, "that person's or organization's information."

Example:

Please collect that person or organization's information on the appropriate form.

-or-

Please collect that person's or organization's information on the appropriate form.

I have looked and looked on the internet at various grammar sites. There are plenty that address the situation when there's a compound possession (i.e., when "and" is used), but none I can find that address the situation when it's not but rather is the possession of one or the other.

On this site, I felt hope when I found the following question:

Using possessive apostrophe with "or"

But then I saw it was wrongly closed as off-topic, saying it should be for the English Learner's site. I'm not an English Learner. It's a legitimate question.

Now, that question does offer one answer, but that answer is obviously wrong because it calls "John or Mary" a noun phrase, which is a phrase that serves as a noun, when it is not a noun phrase but is two distinct nouns conjoined by "or."

When we say "John and Mary's," that's compound possession, which indicates that John and Mary collectively share whatever it is, but when we say "John's and Mary's" that's not compound possession as it indicates that John and Mary each own the ensuing noun, so if the ensuing noun were "information," then it would refer to John's information and Mary's information, like John's address and Mary's address, which may not be the same.

From this, we know that the presence of a conjunction doesn't necessarily indicate compound possession, and when the conjunction is "or," it certainly seems that it wouldn't because "or" specifically indicates one or the other and not both, but therein lies the rub because it makes sense in my head to say "John's or Mary's" or "that person's or organization's," but what sounds right to my ear is "John or Mary's" and "that person or organization's," which is why I'm seeking an answer, an answer I haven't been able to find and hope someone here can definitively provide with some grammar source material that specifically addresses the possessive case with "or" rather than only addressing "and."

Answer

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.