Why Do The Wires Inside a Headphone Cable Touch? Can't Wires Not Touch?
ONE of the conductors can be uninsulated.  It takes TWO uninsulated wires touching to make a short.  In some cases the ground wire is uninsulated to save cost, bulk, and keep the cable as flexible as possible.

Some insulation is very thin and looks like the color of bare copper.  Your "right headphone" example is most likely one of those.  You can check this easily with a ohmmeter or continuity tester.  Hold the probes a few mm apart on the wire.  If it is insulated, there will be no connection (infinite resistance).