Barrel nut contact?

Does the barrel nut need to be clearanced from the stock on the sides?

I would say the less unnecessary stress the better. I can't claim to Know if it will make a huge difference but it certainly is unnecessary to leave a pressure point.
While it might not be terrible, It's not a good thing.
I would relieve it so it was beyond question. Supported typically is not bad but stressed, pinched or pried on is Never a good thing.

I tend to cut the stock for clearance around the barrel nut.

It is best to have the action touching the stock, pillars, or bedding block, only under the action itself, especially in the area of the action bolts. Not floating the tang "can" cause binding and that can also be true in the area of the barrel nut. That being said, there are some who have found accuracy by bedding 2 inches or so up the barrel, in the same vain that some find it useful to have a "pressure point" in the front of the stock. But if this is done, and the barrel is changed, or the nut is moved, the bedding under the nut has to be ground out, where the bedding under the action does not.

I have never found it necessary to bed under the nut, and only see potential for problems and extra work.

JMHO :-)) .... Good Luck ... Jim