Allow notifications when the app first asks. If you said no, turn them on in your phone's settings: Settings → Notifications → CalmHandoff on iPhone, or Settings → Apps → CalmHandoff → Notifications on Android.
Then check the app's own Settings → Notifications for which kinds you want.
What a notification shows
That something happened — a message, a call, a change to the calendar. Never the contents. Someone reading your lock screen learns that your co-parent wrote to you, not what they said.
- Calls need notifications on, or the app open, to ring.
- Focus and do-not-disturb modes silence them like any other app.