Lyyn ☮️🦄 #2024-02-16, Mastodon's push notifications work by sending the data in the notification itself, encrypted by your server. This makes it more reliable than just a ping that "something happened" that would make the app do an API request to retrieve the actual notification.
Unfortunately, you can't sidestep the need for a centralized notification service if you care about battery life, which I imagine most people do. So end-to-end encryption will have to do.