Once you're north of Bishop, 395 climbs the Sherwin grade and you're at a fairly high elevation. I always carry chains and drive an AWD, but I've only had to actually put on the chains once (when I didn't have an AWD).
However, my recollection (when approaching from the south) is that the signs usually indicate the road is clear and no chains are required until Lee Vining or Bridgeport, at which point chain control comes into play.
A good resource is the website for Mammoth Mountain Resort and especially the forums:
http://forums.mammothmountain.com/
In fact, there's one one that is specifically dedicated to highway conditions. You will find some very good advice there.
Here's a good post from earlier in the year about the drive north of Mammoth in bad weather. It seems the poster has the same impression that I do - CalTrans keeps US395 south of Mammoth clear more than they do for the road north of Mammoth:
http://forums.mammothmountain.com/vi...hp?f=8&t=12568