Description of the Problem
When you send an email message from a shared or delegated mailbox in Outlook, the sent message is not saved in the Sent Items folder of the shared or delegated mailbox. In other words, if a message is sent from within one's personal account by changing the "From" address to a shared or delegated account, the sent message is saved in the sender's personal Sent Items folder. This may be a problem for shared mailbox users, as the shared account Sent Items folder cannot be relied upon for accurate record-keeping of sent emails sent from that account; and multiple users with access to the shared account will not see messages sent from that account if it is sent by other shared account users because it is stored in the sender's personal Sent Items folder. A similar issue can occur when a delegate deletes an email from a shared or delegated mailbox. The deleted items can end up in the delegate's personal deleted items folder.
Messages sent from a shared account using OWA are properly saved in the Sent Items folder.
See Microsoft Support Article for sent items: https://support.microsoft.com/en-us/help/2843677/messages-sent-from-a-shared-mailbox-aren-t-saved-to-the-sent-items-folder-of-the-shared-mailbox-in-outlook
See Microsoft Support Article for deleted items: https://support.microsoft.com/en-us/help/202517/items-that-are-deleted-from-a-shared-mailbox-go-to-the-wrong-folder-in-outlook