When Outlook 2013 can’t find the correct folder for the Sent Items or Deleted Items folder, it will create a folder within your mailbox which doesn't sync with the mail server. This is why the folder holds “This computer only” behind its name.
To fix this you will need to set the personal folders as the default rather than import into the 'this computer only' folders.
Go to File, Account Settings, Data Files and select the Personal folder file and set it as default.
When you set the “Root folder path” option in your account settings, Outlook will usually recognise the correct folders.
You can root your mailbox in the following way:
File >> Account Settings >> Account Settings >> Select your IMAP account >> More Options >> Advanced >> Root folder path
Usually setting this to “Inbox” will do the trick.
If Outlook still didn't pick up the Sent Items folder even after restarting Outlook, locate your OST-file for the IMAP account and rename it to .old while Outlook is closed but make sure you backup any folders marked with “This computer only” first.
To determine the location of the OST-file use:
File >> Account Settings >> Account Settings >> Data Files
Select the data file of the IMAP account and press: Open File Location
Upon restarting Outlook, a new OST-file is automatically created and the mailbox is being re-cached. Outlook will then try to determine the correct Sent Items folder again.
With some IMAP servers, Outlook 2013 will still have issues determining the correct folders. Sadly, there is nothing you can do in this case other than reporting it as a bug via Microsoft Product Support.