Speaking as an American, I usually just say "driver side" and "passenger side".
Honestly, until today, I never heard "near /off" terminology used for cars! I ride horses, and up 'til now, have only heard "near side" in reference to the left (or mounting) side of the horse; and "off" as the right side (opposite from the mounting side).
Using "near/off" for cars would confuse me, because I have it so ingrained that "near" is always the left side. (You know, like the markings on the rabbits in "Watership Down"! The "Off Hind Mark", etc. <G>)