It is not called the ‘Private Share’ and that is by design. WD decided that a User Private Space would not be accessible from the LAN in the ‘Home’ setting. If you wish to share over the LAN, put those files in the Public SMB or CIFS share for the best performance.
Any sharing of the Private Space should be done through WD Discovery program (desktop, web app or mobile app). If that is buggy as you indicated and it could well be with the ever changing macOS buggy file system, you should look for support with WD technical at https://support-en.wd.com/app/ask and provide your macOS version and debug_logs.
