What to do with wire shielding before attaching the plug?
Yeah, it depends on the application and if your device(s) are in a high noise environment (machinery,high switching noise from components, etc.). Also, are you sending data down this USB line? 

But to answer your main question, this shield would normally go to a chassis ground, on both ends. If connected to another ground, it would cause a ground loop which, in practice, aren't ideal. These can be hazardous in some situations. Eliminate multiple grounds when possible.