Yes, you have a choice right now, if you have the “My Contacts” enabled on your account, you can send notifications in smartapps with the UI to specific contacts. Without it, notifications go to all contacts.
However, the built in notifications like Hub offline, low battery and SHM that have no UI for selecting the contacts are broken and will not send out notifications.
@tyler helped me with a workaround but I don’t know if it should be made public. Hopefully ST will fix the built in Smartapps and expose the ability to choose who should get notifications.
Otherwise the only real answer is have support remove the “My Contacts” / contact book from your account and those notifications will start again. However, any custom notification smartapps will then break.