Is there a way to share information not found in public records specific to a deceased individual?
Yes! There are several methods. One popular method is to upload scans of documents to the Memories page of a person profile, then attach them on the Sources page.
See for example Elder John Crandall LZGV-W4V: Sources, Collaborate, Memories all are heavily used.
The Help Center, under the ? in a circle in the upper left of your web interface on FamilySearch, has lots of information about how FamilyTree works.