In practice, most arm cables seem to keep the two channels separate, but that may be partly because it's easier to do it that way with coaxial or twin-core-and-screen cables. The left and right channels run together inside the arm, of course.
There's no need to screen the ground lug cable, in fact I've never heard of anyone doing so. If you think about it, there's no signal to screen
To minimize capacitive cross-talk in particular with MM which have a significant and increasing source impedance with frequency you'd want to keep the signal leads for each channel separated so you would shield each individually. In the case of moving coils its much less critical because of their much lower source impedance.
Because of mass and flexibility considerations it is very rare (I've never seen it) to shield the wiring in the tone arm itself - hopefully the grounded (!) metal arm tube takes care of that. Some people twist the channel pairs which certainly can't hurt but is more effort than I've been willing to put into it..
No need to ever shield the ground cable, as mark pointed out there is no signal to be shielded.