[beering]

TobTobXX is correct about how to do push notifs without transmitting the message through Google or Apple servers. This lets the app can fetch the message through its own channels (and you should be able to inspect the push payload coming from Google to verify this).

Once the messaging app has downloaded the message securely, the "end" in "end-to-end" is satisfied and what you do with your notification is your decision. If you elect to explicitly grant Pushbullet permission to read your notifs and forward them to your desktop, then that's your choice I guess.