This Cmdlet manage the right properties
Set-Mailbox –Identity 'sharedmbx' –MessageCopyForSendAsEnabled $True –MessageCopyForSendOnBehalfEnabled $True
The MessageCopyForSendAsEnabled property is False by default. If set to True , Exchange Online keeps messages sent by delegates as the shared mailbox in the delegate's mailbox and also creates a copy in the shared mailbox. The same about MessageCopyForSendOnBehalfEnabled property.