All push notifications require an app to receive that notification. For now, we're offering integration with Prowl. If you use an alternate app to receive push notifications from third party services, let us know, and we can see about adding integration!
Most email filters stop processing as soon as they match a filter. For push notifications, we made them work like subject tagging. So, if a push filter matches, it will notify you, but continue checking your other filters in case you want to redirect or CC it to another person.
In order to keep an overly-aggressive filter from going off all the time, push notifications are limited to one every 5 minutes. You will need the Prowl app (again, if you're using another service like Pushover or Notify My Android, please let us know) to receive push notifications. Check out our help page on setting up push, and start receiving notifications today!