A private group is a type of group you create for a 1-on-1 client.
Private groups do not come with discussion forums, group contests, or collaboration features since it's meant for just 1 client. All the other features remain the same as in non-private groups.
PLEASE NOTE: You can have more than 1 client in private groups, but they won't be able to see or communicate with each other.
Another purpose of marking a group private is for convenience, allowing you to only show private groups. This is great if when you have a lot of groups and want to easily see the groups for your 1-on-1 clients.