This solution may generate a storm of criticisms but may also give you an idea of a path to follow to get where you want.  Note that this solution works perfectly in Mathematica 9 but in Mathematica 10 there is a problem that is preventing the last Graphics object added to copy to the right side with the right colors.  

    Grid[{{
       Labeled[Button["Copy to Right Side",
         Block[{cellContents = NotebookRead[EvaluationCell[]]},
          rightSide = 
           cellContents[[Sequence @@ 
               Position[cellContents, GridBox[{{_, _}}, ___]][[2]]]][[1, 
             1, 1, 1, 2]];
          rightSide = 
           ReplaceAll[DisplayForm@rightSide, GraphicsBox -> Graphics];
          ]
         , ImageSize -> Automatic
         ], Style[g, Deployed -> False]]
       ,
       Labeled[
        "Copy of Left Side"
        , Dynamic@rightSide
        ]
       }}
     , Alignment -> Left
     ]

This is how it looks after the first evaluation:

![enter image description here][1]

And this is how it looks after adding two Graphics elements and pressing the "Copy to Right Side" button.

![enter image description here][2]


  [1]: https://i.sstatic.net/yxdiE.jpg
  [2]: https://i.sstatic.net/k9RL3.jpg