Dreams stores all data locally on your device.
If you are logged into iCloud and have enabled its backup option or the Dreams iCloud sync feature, your data will be backed up to your personal iCloud account.
You may opt to integrate Dreams with your personal Apple Health app database. If you grant Dreams access to this database, the app will save the bedtime and wake up time to Apple's Health app database.
Dreams cannot read from or write to your Apple Health app database unless you give it explicit permission to do so.