When using the Rest API to query for list of Groups, you can get the group type (secrete, member only, public, etc). But when querying for spaces, I don't get any field stating if the space is private or public. Is there another call that can be made to get this detail?
Reviewing through your inquiry, the two container types (spaces/groups) are not comparable in this way as spaces are controlled via user group permissions and do not have "public/private" types, whereas social groups do.
For spaces, I would suggest checking out the Entitlements API as this would allow you to pull a specific user's or user group's entitlements on a specific space.