So The object on the left has a full weight and it moves slower than the bone when they are both moved. The second mesh has a weight of 0 for the bone on the left and does not move like it's supposed to. The mesh on the right has a full weight for its bone (right bone) and does not move when its bone is moved, yet the mesh on the left does while having a weight of 0.
I tried removing the armature modifier on both meshes (as you can see, one still does not have the modifier) and re-applying them, but the same issue is occurring. I even deleted the bones, created new ones, re-positioned them and still a fail.
Also, there are two spheres that contain these objects and they move the same speed as the bones do when moved, and don't move the one on the other side.
EDIT: Here is the .blend file.