During onboarding, you'll be asked if you'd like to enable notifications. For example, Cocoon will send notifications when people reply to your posts. If you'd like to receive these, you'll need to enable notifications on this screen.
One important note: Cocoon only sends "quiet notifications", which means that they do not make any noise or vibrate your device. The intention is to reduce the potential for interruption, while still ensuring you see when there is new activity in your Cocoon.
You can control exactly which notifications are sent on the Settings tab.