Send emails are not saved to Office 365 / Exchange shared mailbox sent items
Emails sent from a shared mailbox (users have full access and send as permissions) are saved only in user's sent items folder, but not in shared mailbox sent items.
Connect of Office 365 via PowerShell and run following command:
set-mailbox <mailbox name> -MessageCopyForSentAsEnabled $True
To check status of the MessageCopyForSentAsEnabled property:
Get-Mailbox Sharedmailbox1 | select MessageCopyForSentAsEnabled