Is it possible to subscribe to notifications when surfing in incognito mode or with private browsing (e.g. on Chrome, Firefox)?
Unfortunately it's not possible to use web push notifications or subscribe the user to notifications when the user is surfing in incognito mode.
The problem is that incognito mode is made to avoid that data and other traces are saved from the website on the current device. Unfortunately the notifications would need to store a Service Worker and memorize the push subscription on the current device in order to work.
If you are browsing a website in incognito mode the permission for notifications is set on "denied" (instead of "default"). This is equivalent to a user that clicks "Block" on the permission prompt... however in this case the prompt is not even displayed to the user and "denied" is returned immediately to the website.
On Chrome, if you click the lock near the address bar in order to manually allow the notifications, you will see this message:
Notifications: Not allowed in Incognito
The fact that notifications are not allowed in incognito, doesn't mean that they are bad for privacy: they are, in fact, a well-designed technology from a privacy perspective.