By group privacy, I mean people not part of the group would be unable to look at the discussions/documents and it would be invite only. Also, would it be possible to have the group be 'invisible' or unsearchable?
Sounds like you should explore a Private or Secret group.
Membership is open and non-members can view content and participate.
Membership is open and non-members can view content, but must join to participate.
Membership is by approval/invitation only and only members can view content and participate.
Membership is by invitation only, only members may participate, and the group is not listed in the group directory.