Messages that are sent by using the "Send As" and "Send on behalf" permissions are copied only to the Sent Items folder of the sender

UPDATED 21-08-2020 - Added Exchange Online method

This is something I didn't know existed but keeps popping up as a frequent question in our organization.

An outlook user with access to a shared mailbox, send as or send on behalf of rights sends an email from the shared mailbox, but the sent email doesn't get saved in the shared mailbox sent items folder but in the users sent items folder.

Since Exchange server 2010 SP2 update roll-up 4 it's possible to specify where the sent emails will be saved. This also goes for Exchange server 2013 CU9 and up.

First check the current settings for a shared mailbox:

Get-MailboxSentItemsConfiguration -Identity sharedmailboxname            
            
RunspaceId                  : 2d911aef-3416-42f9-9900-531d0fcdea94            
SendAsItemsCopiedTo         : Sender            
SendOnBehalfOfItemsCopiedTo : Sender            
Identity                    :            
IsValid                     : True

As you can see the "SendasItemsCopiedTo" and "SendOnBehalfOfItemsCopiedTo" is set to "Sender"
There are 2 settings to choose from: "Sender" "SenderAndFrom".

To set it to save in both mailboxes use:

Set-MailboxSentItemsConfiguration -Identity sharedmailboxname -SendAsItemsCopiedTo `
senderandfrom -SendOnBehalfOfItemsCopiedTo SenderAndFrom

Check to be sure:

Get-MailboxSentItemsConfiguration -Identity sharedmailboxname            
            
RunspaceId                  : 2d911aef-3416-42f9-9900-531d0fcdea94            
SendAsItemsCopiedTo         : SenderAndFrom            
SendOnBehalfOfItemsCopiedTo : SenderAndFrom            
Identity                    :            
IsValid                     : True

Set it to all shared mailboxes in a nice one liner for Exchange 2010:

Get-Mailbox -ResultSize Unlimited -RecipientTypeDetails sharedmailbox | Set-MailboxSentItemsConfiguration `
-SendAsItemsCopiedTo SenderandFrom -SendOnBehalfOfItemsCopiedTo SenderAndFrom

Set it to all shared mailboxes in a nice one liner for Exchange 2013:

Get-Mailbox -ResultSize Unlimited -RecipientTypeDetails sharedmailbox | Set-Mailbox `
-MessageCopyForSendOnBehalfEnabled $True -MessageCopyForSentAsEnabled $True

For Exchange Online there are two ways of doing this, in is in the Office 365 Admin portal and the other is in PowerShell.

In the Portal go to Groups - Shared Mailboxes

Then find the Shared mailbox you want to edit and edit "Sent Items"

And here you will see the two toggles to choose you options

With PowerShell:

After loggin in to Exchange Online Check the shared mailbox you want to modify:

Get-EXOMailbox -Identity sharedmailbox@domain.com | FL

These are the attributes that we triggered from the GUI toggle within the Office365 Admin Center earlier:

MessageCopyForSentAsEnabled : True
MessageCopyForSendOnBehalfEnabled : True

Set these attributes for a single mailbox:

Set-EXOMailbox -Identity sharedmailbox@domain.com -MessageCopyForSendOnBehalfEnabled $true -MessageCopyForSentAsEnabled $true

Enable these settings on all your shared mailboxes in your environment:

Get-EXOMailbox -RecipientTypeDetails shared | Where-Object {$_.messagecopyforsentasenabled -eq "" }

If you have to return all shared mailboxes that has the Sent Items Copy enabled:
Get-EXOMailbox -RecipientTypeDetails shared | Where-Object {$_.messagecopyforsentasenabled -eq "true" }

Take the selection of the shared mailboxes that don’t have the Sent Items Copy enabled yet and combine that with the Set-Mailbox command that enables both Sent Items Copy attributes:

Get-EXOMailbox -RecipientTypeDetails shared | Where-Object{$_.messagecopyforsentasenabled -eq ""} | Set-EXOMailbox -MessageCopyForSendOnBehalfEnabled $true -MessageCopyForSentAsEnabled $true