Short Answer: Currently, NOTHING; and, NO ONE - but, there are 'caveats'.
The 'Status' of YOUR personal "FamilySearch Person Identifier" (ie. PID), that being you as a User/Patron of 'FamilySearch'; as, a (CURRENTLY) "Living" individual/person in "Family Tree" of 'FamilySearch', will most likely NOT be "Changed" that of "Deceased", when you Die; UNLESS, you are a Member of the Church; and, your 'Ward/Branch Clerk' changes you status in the Membership Records of the Church to that of being "Dead".
IF, you are not a member of the Church; THEN, members of your immediate family, both, (1) are aware of your (personal) "FamilySearch Person Identifier" (ie. PID) in 'FamilySearch'; and, (2) formally, notify (with 'Proof'/'Evidence') 'FamilySearch' of your Death.
That said ...
My understanding is that 'FamilySearch' is EXPLORING avenues with regard to, "Inheritance"; and/or, "Bequeathing", of the "Living" (Users/Patrons) individuals/persons (and, most likely, the "Memories" of such) in the "Private Spaces", of a recently deceased Family member, to an immediate Family member.
And, I also understanding is that 'FamilySearch' is EXPLORING much MORE in regard to "Living" (Users/Patrons) individuals/persons in our "Private Spaces" ...
Something that WILL take some time, 'make no bones about' that ...
There are a myriad of "Privacy" Laws within the many various Countries and Unions throughout the World - it is a nightmare to negotiate.
"Family Tree" of 'FamilySearch' is used in many Countries and Unions throughout the World; and, as such, must adhere to the myriad of "Privacy" Laws throughout the World.
Hence, "Privacy" is one of the reasons that we cannot "Share" the "Living" individuals/persons in our own "Private Spaces" - it is NOT the ONLY reason; but, certainly has a bearing on the matter.
Another factor is that such would have to be 'Coded'/'Programmed' into the Programme that is "Family Tree" of 'FamilySearch'.
Just my thoughts.
Brett is correct
which is why you should be cautious and judicious in what you really want to mark as private. My personal feelings (which may very well be different than other users) is if it really is super private stuff - then chances are you really shouldn't be uploading it to Family Search in the first place.
also note that if you have items that are only linked to persons but are "public" - people CANNOT do a "search on such items" to find them - UNLESS you share with them the URL of the item or album. This is what I do on my personal items that are linked to living persons - I leave them public - but they would normally only be accessible to people I have given the URL to - which would be close family.
here are some links of possible interest (noting that these writeups were made BEFORE the private option became available - which was not very long ago)
again I emphasize - these are MY opinions and feelings and there surely are people with opposing opinions about the use of private/public options. and I respect these differing opinions.
it all boils down to how secure you really want to be. the PRVATE option is very secure. but the drawback is what happens when you die and no one has access to your account or your privatized items.