Office 365 and Exchange 2013 shares the same behaviour when you speak about Send As and Full Access permission to other mailboxes than yours.
Giving these access permissions to a mailbox and open it on Outlook or OWA when you send an email as the shared mailbox or the other mailbox you have the right to you will find that the email you have sent will be moved to the Sent folder of your own mailbox and not to the the Sent folder of the Shared Mailbox.
This behaviour could be changed on Exchange 2010 via powershell. In Office 365 / Exchange 2013 this cannot be changed.
For Outlook (2007 / 2010 / 2013) there is a fix : having the Outlook fully patched you need to add this line to the registry (change the 15.0 with the right number – 15.0 for Outlook 2013, 14.0 for 2010 and 12.0 for 2007) :
HKEY_CURRENT_USER\Software\Microsoft\Office\15.0\Outlook\Preferences\DelegateSentItemsStyle = 1
Email deleted from the Shared Mailbox will be moved to the Delete folder of your own Mailbox.
For OWA, there is no way to let the server to put the email in the right Mailbox’ folder. One thing you can do is open the other Mailbox using the menu and selecting the Open another Mailbox option: in this way you will open another tab or window with the other Mailbox and from here the email will be moved to the correct folder.