When a user presses the Submit button, the account owner (the email listed for the account) will get an email asking for approval for sharing to the community.
If you don't press the Approve button, then that drawing+video will never be shared. Nothing your child shares to the Osmo community will actually go anywhere, unless you approve the drawing or you've set your account to always approve drawings.
Note that this doesn't have anything to do with sharing through email.
To change the permissions/email settings, please go to my.playosmo.com, hover over your account name in the top right corner, select 'Account', then click 'Games' on the left, and look under 'When your child submits a piece of artwork to the Osmo community'. The default value will be 'Email me for permission every time'. If it's not set to that, then someone changed it at some point (probably through a link in an email).
Also, we manually approve all drawings, so nothing your kids submitted would go directly to the community. If there's some specific drawing your kid has submitted that you approved and would like to remove, please send a link to us at email@example.com, and we'll remove it.