Following the Android team's success with their local push notifications, the Growth team is enthusiastic about using push notifications to re-engage lapsed new editors. The Growth features only exist on the desktop and mobile web (not the apps), and so these would be browser notifications.
The use case would be something like sending a notification to newcomers who have completed a "suggested edit" through our features, but then have not edited again for 48 hours. In an initial iteration, this would probably amount to only thousands of notifications per month (not more than 10,000). We would run a controlled experiment to see the impact of the notifications before deciding whether to send to more users and for more reasons.
In the longer term, as discussed on T247438: Estimate push notification volume, we could see the web sending a much higher volume of notifications for power users who want to stay up-to-date in realtime with their mentions or watchlist. But that is not on our minds at this time.
We also know that people have to opt-in to browser notifications, so we would have to think together on the best time and place in a user's workflow to do that.