A delivery group is part of your account, and uses all of its settings -- including password and spam filters. As such, it is typically not appropriate for someone who wants to use it as their primary email address. If you want someone to have a separate account, with their own password, you can add a new user to your billing group.
All mail sent to a Pobox Basic account, no matter how many aliases it has, all goes to the same delivery locations. Pobox Plus and Mailstore accounts can specify different delivery locations for different aliases. A delivery group is a group of aliases that all deliver to the same place. Edit your Delivery Groups now.