Office 365:
Set-Mailbox Shared@domain.com -MessageCopyForSentAsEnabled $True
Exchange:
Import reg file to put sent items in corresponding shared mailbox folder.
Change it according to your Outlook version.
Outlook 2010 = 14.0
Outlook 2013 = 15.0
Outlook 2016 = 16.0
Sent Items:
REG ADD "HKCU\Software\Microsoft\Office\16.0\Outlook\Preferences" /v "DelegateSentItemsStyle" /t REG_DWORD /d 1 /f
Deleted Items:
REG ADD "HKCU\Software\Microsoft\Office\16.0>\Outlook\Options\General" /v "DelegateWastebasketStyle" /t REG_DWORD /d 4 /f