so let me get this, you have two installations of wordpress and your URL for your installation is https://domain.com/public and https://domain.com/private. Are the wordpress installation inside of domain.com or domain.com/public and domain.com/private/

if the installation files are inside of domain.com/public/ and domain.com/private, make sure the MySQL database is separate but can use the same username and password. if you are changing the site address (url) and wordpress address url are the same such as https://domain.com/public for both. and same goes with the private.

I would also recommended instead of doing sub directories for individual installations. why not use sub domains instead…

https://public.domain.com and https://private.domain.com, this way, u won’t have any conflicts