When you set up a Shared Mailbox in Exchange Online, and then assign someone delegate access to it, for some reason the default settings copy the Sent Items from the Shared Mailbox into the sender’s Sent Items, not the Shared Mailbox Sent Items. No idea why, but that’s just how it is.
Anyway, connecting to Exchange Online via PowerShell, there’s a quick and easy way to fix this.
Set-Mailbox -Identity [email protected] -MessageCopyForSendOnBehalfEnabled $true -MessageCopyForSentAsEnabled $true
This will copy Sent Items to the Shared Mailbox no matter if they are Sent As, or Sent on Behalf Of.