Is this for Exchange Online? If so, you can toggle the flag to have messages saved in both the delegate and delegator's mailboxes. Details are here: https://learn.microsoft.com/en-us/exchange/recipients-in-exchange-online/manage-user-mailboxes/automatically-save-sent-items-in-delegator-s-mailbox
If you are using Exchange Server, the behavior can be modified by registry key: https://learn.microsoft.com/en-US/outlook/troubleshoot/email-management/email-remains-in-the-outbox-when-you-use-the-deleg