Personally I am sure that this doesn't answer the question but it seems important.
Assume that the purpose of genealogy is to construct family relationships and potentially add further information about the lives of those people.
As I understand the question, the two people have found sufficient evidence in "paper-based" sources that they have a common ancestor. Also they have identified that they share a small amount of DNA. Note that the identification of the common ancestor does not depend on the DNA (though it might have been a spur for investigation).
The question appears to be whether the independently proven common ancestor and the DNA match can be used to deduce that the matching DNA came from the common ancestor?
I would like to query the value of concluding that the matching DNA comes from the common ancestor. As far as the two researchers go, the "paper" evidence has already demonstrated the relationship. Deciding exactly where that matching DNA comes from would therefore seem to only have relevance if a third researcher appears.
If the third matches both the first and second researchers, there is no immediate reason to assume that the third matches the first and second on the same bit of DNA, so no reason to assume that the 3rd shares the same common ancestor.
If the matching bits of DNA can be identified and the 3rd matches both 1st and 2nd on the same bits of DNA as the 1st and 2nd match, then several possibilities hold.
The 3rd matches both by chance. (I have no idea of the probabilities of that).
The 3rd matches both by descent from the previously identified common ancestor.
The 3rd matches both by descent from an ancestor of the previously identified common ancestor and it is purely by chance that the common DNA has mutated in the 3rd to match. Again, I have no idea of the probabilities.
As I said, this does not answer the original question - rather it queries the value of answering it, especially when only the 2 researchers are involved.