For emails sent as from the shared inbox, make a copy of the email in the shared inbox's sent items folder as well.
set-mailbox <mailbox name> -MessageCopyForSentAsEnabled $True
See this TechNet article for more info.
First remove the permission for the user:
Remove-MailboxPermission -Identity sharedbox@internet.com -User allan.jude -AccessRights FullAccess
Then give the user permissions again with AutoMapping turned off this time.
Add-MailboxPermission -Identity sharedbox@internet.com -User allan.jude -AccessRights FullAccess -InheritanceType All -AutoMapping $false
Now you will have to manually add it to the user's Outlook. This will give them access to view emails with the Private flag.
More info can be found on this TechNet article.
Another option I've heard of is to make a transport rule that changes private emails to normal. Might look into that in the future if there continues to be issues.