When you send a message from the shared mailbox, the sent message goes into your own Sent Items folder. By default, sent messages are stored in your own mailbox, not the shared mailbox. Previously, the solution was to edit DelegateSentItemsStyle registry value but most of end-users cannot complete this request or even create more problems. Now, there is an update, allowing customers to setup this option very easily.
You should access Office 365 tenant area and select the shared mailbox you wish to edit. Simply enable the following options: