Shared mailboxes are mailboxes that are used by more than one person by nature. It appears as a second mailbox in users’ Outlook and you can check incoming mail from there. However, sent e-mails only appear in the sent items section of your mailbox. This prevents other users using the shared mailbox from being notified of sent e-mails.
You can also save a copy of the sent items to the shared mailbox by making a change from the M365 Admin Center.
After logging into M365 Admin Center, follow the Teams & Groups>> Shared mailboxes menu. When you click on the shared mailbox, a menu will open on the right of the screen.
From this menu, click the edit button in the Sent Items heading. Check the Copy items sent as this mailbox and Copy items sent on behalf of this mailbox options and save.
Now, when you send mail on behalf of the shared mailbox, you will see a copy of the items drop into the shared mailbox sent items folder.