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:

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

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.

    g = Graphics[{Yellow, Disk[]}, ImageSize -> 200];
rightSide = Style["Left side will be copied here.", Gray, Italic];
Deploy@Panel@
  Grid[{{
     Labeled[Button["Copy to Right Side",
       Block[{cellContents = NotebookRead[EvaluationCell[]], rs},
        rs = 
         cellContents[[Sequence @@ 
             Position[cellContents, GridBox[{{_, _}}, ___]][[2]]]][[1,
            1, 1, 1, 2]];
        rightSide = 
         ReplaceAll[DisplayForm@rs, 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:

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