The original object on the right is after applying a proportional transform to the face of a cylinder. The object on the left is a duplicate of the original showing the distorted mirror effect. Mirroring was set up with "Symmetrize" and selecting "X Mirror" on the "Mesh Options" tab.
The problem is that the X-Mirror option of the Mesh Options tab works by editing the mirrored vertices only when there is an exact X-mirrored counterpart (in local space).
Symmetrize does indeed create a copy on the other half and, in principle, if you are very careful, you can work fine with that X-Mirror option enabled. The problem is that if you accidentally duplicate vertices only one of them will be moved along with the mirrored one, the other one will remain static, which is what appears to have happened in your case.
As you can see, this is not a very good way of working with a symmetric mesh. Another complication arises when you want to add new vertices. You need to symmetrize every time so the X-Mirror option has mirrored vertices to work with.
There is a much better way of working with symmetric meshes, which is with the Mirror modifier, which works by copying every vertex from your mesh on the other half (in local space). At every time, you have an exact copy on the other half. Adding vertices or moving them also adds them or moves them on the other half. No need to use symmetrize. When you have the finished model you can click Apply on the modifier to make the changes permanent. At that point, the mesh becomes one and the mirroring stops working.
When you wish to add details that shouldn't be mirrored use a new object. Since modifiers are attached to the object, a new object without the Mirror modifier won't be mirrored.
Here is a .blend with the Mirror modifier used on your mesh. I used a separate layer so make sure to be on the correct layer. Notice that the X-Mirror option is disabled now. Also, I left the bridge unconnected so you could understand why you should check the Clipping option of the Mirror modifier. To see that, just move the edge loop of the bridge so it meets with its mirrored edge loop.
If you want to see the mirrored mesh in "edit mode" too, click on the last of the icons showing on the Mirror modifier settings. You can hide the mirrored mesh too if you click on the eye icon.
As an aside, the arm had duplicate vertices. While you are editing, make sure to select the part of the mesh you are working on and press W and Remove Doubles to not end up with duplicate vertices.
Also, to get better results on the curvature of the arm, try editing by extruding edge loops instead of using proportional editing. The basic workflow is by using an orthogonal view, pressing E to extrude the edge loop, then moving it where you want it, then pressing R to rotate it and S to scale it if necessary. If you are done and the result is not smooth enough, you can apply a Subdivision Surface modifier.