Currently (19.11), the Upload (Any File) tool does not let you change the parameter for Public. There should be a way for editing uploaded files into public and vice versa. Currently, if you want to do this, you have to hack the database and change the 'public' column to 1 if it's public and remove the value '1' if you don't want to make it public.
*** This bug has been marked as a duplicate of bug 28098 ***