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Is everyone logging into their own Spotify account when accepting the invite to join the family plan?
The way Spotify Family works is that each person creates their own Spotify account first if they do not already have one. For everyone to create a Spotify account they will need their own email address.
Once everyone has their own Spotify account you then invite them to join your family plan and then they sign in with that account in the app. Make sure everyone enters the same home address when it asks for verification. Then everyone will be able to listen on their own without interrupting the other users.
Check out these pages for help with the Family Plan: