Definitely, but they don't have to contain any (plaintext) message content for encrypted messengers.
On Android, push notifications were always processed by the receiving app, so it can just decrypt a payload directly (or download new messages from the server and decrypt these); on iOS, this isn't as reliable (e.g. swiping the app out of the app switcher used to break it in several iOS versions), but "VoIP notifications" and the newer "message decryption extension" [1] are.
The same principle applies to Web Push – I believe end-to-end encryption is even mandatory there.
On Android, push notifications were always processed by the receiving app, so it can just decrypt a payload directly (or download new messages from the server and decrypt these); on iOS, this isn't as reliable (e.g. swiping the app out of the app switcher used to break it in several iOS versions), but "VoIP notifications" and the newer "message decryption extension" [1] are.
The same principle applies to Web Push – I believe end-to-end encryption is even mandatory there.
[1] https://developer.apple.com/documentation/usernotifications/...