I'm not sure I fully understand you, but I've never seen a person's NAME replaced by "Living" in either FTM or Ancestry, whether or not they're private. A private entry, typically someone who is living and in another tree to which you don't have permission to see living individuals, shows as simply "Private" with no name or dates at all, but that's only online, not in your local FTM tree (you can't download someone else's tree to FTM even if you have access to it, so there shouldn't ever be any notion of "Private" locally).
The word "Living" is a separate marker that may replace the date of death, but doesn't itself affect the name. Normally, "Living" applies to anyone without a death fact and born less than 100 years ago, but you can override that manually (although there was a temporary bug on Ancestry a year or two ago where everyone without a death fact was being set to living regardless! [edit: here's a discussion about it:
https://www.ancestry.com/boards/topics.software.famtreemaker...])
If you've updated someone online from a match and that's set them to "Living", then you've synced to FTM, I'd expect their death-date field in FTM (NOT their first name) to show "Living" - that's correct and how it should work. [edit: I seem to be mistaken! - I can't see any "Living" indicator in FTM at all, even after syncing a living individual from Ancestry - see attached screenshots]
Perhaps you could post screenshots of the person, both online in Ancestry and in FTM, showing where the word "Living" is appearing? That might help us understand the problem.
[edit: added screenshots showing a private individual from another tree in Ancestry, a living individual in my Ancestry tree, and the same individual in FTM after syncing]