No special changes are required to make Spaces work in Private, unless it is in conjunction with an external application or something.
Just click on the three vertical dots, select Settings, and press Make private.
This will make it a Private space.
If you press Make public, it will become a Public space.
If it does not work, then you have another problem that has nothing to do with Private.
P.S.
This worked fine when I duplicated it in Private. Maybe you just had a bad build. Sometimes there are glitches.
Also, Spaces and models that are set to Private are visible as 404 to everyone but you, unless you use a token. That’s what this feature is for.