You do have weight painting/vertex groups assigned to your mesh .
It looks like during the parenting process you must have chosen bone heat to add vertex groups/weights to the mesh and It looks like when you mirrored the armature and/or fixed the naming issues later the weights somehow got screwed up .
The easiest way to fix this is to delete the half of the mesh with the problem and then to add a mirror modifier to the mesh . Just make sure to move the mirror modifier to the top of the modifier stack above the armature modifier stack so that the mesh moves as expected when posed even when you cross the mirror axis . You can apply the modifier if you want, but it isn’t necessary and you can continue editing the mesh if desired .
There was(is?) a script to copy weights to left/right sides of a mesh in the 2.4X series, but since I never used it and since i don’t know where it is or if it still works with 2.5X series, I can’t be of much help with that . You could try and do a search for that option if you are interested .
Also you might want to parent the IK target bones in your rig to a bone beyond the IK chain length to make them move with the mesh when bones in the body are posed like I have done in the attached blend . I just parented the arm IK targets to the the chest bone and left the legs alone .
character7FIX.blend (582 KB)