Should an app push notification system send 'critical, system notifications' if a user has turned off push notifications (much like email systems do). Bearing in mind that contact via email will always be a valid way to communicate critical system communications. An example of a critical system message would be 'Your account is about to expire'.
If the user has opted out of push notifications, it violates the user's trust in your system to override their actions. If their account is about to expire you have two other ways to communicate with said user.
- Send the notification via email as you have listed above.
- Display a banner/warning/tip within the application itself.
Often, users have opted out of push notifications due to the overuse/misuse of push notifications and pushing them further on the user after the user has actioned out of receiving them will serve to further upset said user.
What a system finds "critical" and what a user finds "critical" are not always the same. A user may have disabled notifications for many reasons. The system's importance may be lower than it thinks it is. The user might not be using the system anymore. The user might be on vacation.
One approach that could solve this particular problem is to allow thresholds for notifications. That is, the user could unsubscribe from low-priority notifications separately from high-priority notifications. Then the intent is clearly communicated on both sides. The only tricky part is categorizing the notifications appropriately.
I'm not sure account expiry could be considered critical. You'd have many opportunities to communicate such status to an engaged user, a push notification to an unengaged user is unlikely to cut through and obtain the desired action and result.
For both app and web based pushed notifications I'm not sure that once opted out of unless the users opts in at a later time you'd actually be able to send true push notifications as you'd not have the required permissions on the client. Of course notifications don't have to be truly push but these are less relevant as they'd not appear in the notification draw on the device OS.
Truly critical notifications required by regulatory standards or for legal reasons are usually exempt from user specified communication preferences. In these cases I can't see relying solely on a push notification being sufficient. Here an email, SMS or postal letter would be needed to cover the provision of the required information.
I think the term 'critical system notifications' is key here. I presume that this is critical to the system, business and possibly the user in this order of criticalness (i.e not very critical).
If thats indeed the case, and the user has consciously turned off push notifications then I think you should RESPECT the user and try to communicate this message in another way e.g. via e-mail as you suggested. The reason is because even if you save the user and help them out of a potentially bad situation like account expiration, afterwards, they will suspect what your intentions were and whether any of the settings actually work.
They wonder how you managed to notify them despite them having turned notifications off. If they begin to distrust you, they will eventually leave you because if you can violate their settings once, you can violate it again.