The communication problem was largely caused by an major Airbus design flaw: the sticks between the left and right seats aren't linked. In other planes, the pilot would have known the copilot had the stick pulled back because the action would make his own stick move back as well. On AF447, the pilot saw nothing other than the copilot's hand on the stick and assumed he was doing the right thing, and in the understandable confusion as they struggled to gain control of the plane the copilot never verbally corrected the misconception.
Incorrect, there is no design flaw. As designed, if both pilots make a flight control input simultaneously, they will receive an aural warning: "Dual Input". They will know about it and either pilot can take priority over the other by pushing a button on the stick which will lock out the other.