Shared mailbox should keep a copy of sent items by its Member's Mailbox

 I understand that a mail sent from Shared mailbox should keep a copy of all emails sent by all members in its own Sent Items folder.

Note that the new shared mailbox sent items behavior is disabled by default and if Admins want it enabled, they can do so by using CMDlets mentioned below.

Connect office 365 to powershell

Run Windows PowerShell as an Administrator.

Set-ExecutionPolicy RemoteSigned

$cred = Get-Credential 

$ExchangeSession = New-PSSession -ConfigurationName Microsoft.Exchange -ConnectionUri "https://outlook.office365.com/powershell-liveid/" -Credential $credential -Authentication "Basic" -AllowRedirection

Import-PSSession $ExchangeSession

A copy of the sent mail will also reside in the Sent Items folder of the member’s personal mailbox.

Set-Mailbox -Identity [Shared Mailbox Identity] -MessageCopyForSentAsEnabled:$true -MessageCopyForSendOnBehalfEnabled:$true

A copy of the sent mail will not reside in the Sent Items folder of the member’s personal mailbox.

Set-Mailbox -Identity [Shared Mailbox Identity] -MessageCopyForSentAsEnabled:$true -MessageCopyForSendOnBehalfEnabled:$False

You can also refer to the below link for details.

http://blogs.technet.com/b/exchange/archive/2015/03/03/want-more-control-over-sent-items-when-using-shared-mailboxes.aspx