Welcome to the "Community.FamilySearch" Forum.
I am just another 'lowly' User/Patron ...
Just in passing ...
As far as I am aware ...
Short Answer: One CANNOT, have a PRIVATE "Fan Chart" (ie. "Pedigree"), in 'FamilySearch'.
As, 'FamilySearch'. is very much "Public",
I would humbly suggest, that the place to maintain, a PRIVATE "Pedigree", would be on one of the 'On-Line' "Websites" (eg. "Ancestry_com"; or "MyHeritage_com"; or, "FindMyPast_com"; or, the like); and/or, 'standalone' personal (computer) programmes (eg, the OLD, now no longer supported, "PAF"; or, "Ancestral Quest"; or, the like).
I hope, that this may help/assist, somewhat; and, I hope, that this may provide you with, some additional, insight; and, perspective; plus, other avenues to pursue.
You may, want to consider, using one (or, MORE), of the "Third Party" Applications, that is (are) "Certified", to work, with "Family Tree", of 'FamilySearch'.
There are BOTH,
(1) Family Tree Management; and,
(2) Charting (/Printing),
Such, can be found in the "Solutions Gallery", of 'FamilySearch':
Family Tree Management
I hope, that this may also help/assist, somewhat.
I guess it depends what exactly you mean by
"How can I create a private family fan chart?"
For example if I go to my own record in FS and I click on "View Tree" and select type "Fan Chart"
I can see a Fan Chart view that starts with me as the root person (a private person)
You should be able to do the same. . .
I can then "print" and either print a paper copy or print a digital pdf version.
I could then even upload that pdf version to FS Memories. (But I would mark it as "private" since it has info on living persons)
so help us better understand, though, what are you really trying to accomplish?
help us understand the "context" of your question.