I understand you're looking for a way to enable search for project content in "Private" social groups. Unfortunately, the entitlements that control a user's view access to content also controls their search access as well, so this cannot be configured out of the box. I see that you inquired about "Members Only" social group, but I don't believe this will work in your use case, since content here is visible to non-members (the caveat is that in order to participate, you must join the group).
To workaround this, I might recommend creating a visible document (e.g. in a public place) that links to the other private content. Users who click on the links will then be prompted to request access to the private group.
Let me know if you had any other questions on this.