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FBI Extracts Suspect’s Deleted Signal Messages Saved in iPhone Notification Database
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It depends on the app. Some apps do (or can be configured to) indeed send "empty"/blank notifications which just notify you that you've received a new message from an app, but not from whom, or what the message contains.
However most apps by default will contain more data, such as who the message is from, and some/all of the sent message body.
If you get a push notification on your phone, everything you see in that notification must by definition pass through the push notification service.