I didn't get how the parties knows about new incoming messages. The app has a wallet that watches the blockchain for new transactions involving the user? What i

I didn't get how the parties knows about new incoming messages. The app has a wallet that watches the blockchain for new transactions involving the user? What if the app is turned off? Or you guys have a central server that watches the blockchain and then updates users? In the case of a central server, does that mean this can be taken down if authorities take over the central server?