By default this is not possible.
But you can define an own portal-type for the Member-Folder, configure it as the member folder and give this folder an own workflow with specific permissions for a to be created new role. Give the enabled groups this new role.
Afterwards all members will get a Member-Folder, but only those in the groups(s) with the new role can access them.
This is all just some XML configuration in your profile: Types, Workflows, Rolemap and the settings for the member-folder type. Even through the web this might be doable (not sure about the new role).
One last thing: Member-Folders are created at the first login. Existing Members would not get the new one. There a migration step would be needed.