I'm not sure if I understand the question properly: "private" as we (coppermine devs) use it applies to the accessibility (visibility) of albums. You can grant the permission to view an album to everybody (public), the owner of that album or to a specific group, using the field "Album can be viewed by" on the album properties page.
Coppermine does not come with an option to specify only a specific group to upload to a particular album if you allow several usergroups to upload in the first place. However, you can enable the permission to upload to an admin-created album. Additionally, users can only upload to albums they're allowed to see in the first place.
This being said, I can't see your issue: as far as I understand, you have a particular group (let's name them "artists") that is meant to be allowed to view a particular album (let's name it "contributions by artists"). Only members of that specific group are suppossed to be able to access that specific album - no other user who doesn't belong to that group is suppossed to be able to view the album "contributions by artists". Is this correct? If yes, then go to the album properties screen of our album "contributions by artists", and change the dropdown field "Album can be viewed by" to "Members of the 'artists' group" and save your changes. As a result, only group members of the artists group will be able to access that album. Now you want to allow them to upload to that album as well? OK, go to the groups page, make sure that the group "artists" does have the permission to upload to admin-created albums: set "Public albums upload" to "yes" for the artists group. Don't worry about the word "public" there - it is a bit misleading and should actually be named "Upload to admin-created albums". Save your changes. Finally go back to the album properties screen of the album "contributions by artists" and set "Visitors can upload files" to "yes" and save your changes. Don't be afraid that "visitors" means "anybody" - this is not the case. It just means "people belonging to a group that is allowed to upload to admin-created albums as set up on the groups page". You might think "hey, but I have other groups that are allowed to upload to admin-created albums. I don't want them to upload to the album <<contributions by artists>>", but don't worry: we have set visibility to "members of the artists group only" - visitors who are not allowed to see that album are of course not allowed to upload to it.
For the above mentioned steps you don't need additional lmods or plugins - those are core features of coppermine.
HTH
Joachim