Not a "problem," per se. But not an option or function, either. Once you delete them off the handheld, they're gone from the handheld. What they do in Outlook is dependent on how you reconcile messages.
You've obviously figured out the differences, and it sounds like your BB is functioning exactly as it should.
But the answer to your question is "no...you can't do that"...sorry.
Welcome to the fourms.