Settings → Notifications controls what you're told about: messages, calls, calendar changes, expenses, handoff reminders and the evening summary.
Notifications never show content
A notification tells you something happened. It doesn't show what a message says. Someone glancing at your lock screen learns that your co-parent wrote to you, not what they wrote.
If nothing is arriving on your phone
- Allow notifications when the app asks, or turn them on in your phone's Settings → Notifications → CalmHandoff.
- Check the app's own Notifications settings.
- Check you're not in a focus or do-not-disturb mode.
Your notification settings are yours. Your co-parent can't see them or change them.