When you turn on push notifications on a device, we store that device's push subscription (the push service address and encryption keys), its language and device type. A delivery record for each notification (type, time, whether it was shown or opened) is kept for 90 days; the sender field is deleted after 24 hours.
Notifications are relayed by the push service of your browser's vendor (Google, Mozilla, Apple or Microsoft). Their content is end-to-end encrypted so the push service cannot read it, and they never include the text of your messages.
You can turn push notifications off at any time in Settings, or block notifications in your browser's site settings. Signing out also turns them off for that device.