Each address alert is tied to an Org Unit and will only be matched against staff client instances of that Org Unit and its children.
When viewing the address alerts you will only see the alerts associated with the specific org unit selected in the "Context Org Unit" selection box. You won’t see alerts associated with parent org units, so the list of alerts isn’t a list of all alerts that may effect your org unit, only of the ones that you can edit.
The specific permission that controls access to configuring this feature is ADMIN_ADDRESS_ALERT. Local Administrator level users will already have this permission. It is possible for the Local Administrator to grant this permission to other staff.