I think asking a user permission for push notifications isn't strict enough. IMO, I'd require approval by a 3rd party, much like how mobile apps need to go through an app store. As an alternative, I'd implement a rating system.
If we allow any web site to ask to allow push notifications, every time we visit a new web site, it's going to ask us if it can do push notifications. Without some kind of rating system or centralized approval system, push notifications will just be another venue for spam.