I use two string nocks when shooting three under as has been suggested in some older posts. My question is, do you butt the string nocks right up against the arrow nock or do you leave some room. If not up against then how much space do you leave?

Leave about 1/16 to 1/8". This will keep the knock from getting pinched when at full draw.

1/16" at most for me.  I put my arrow on and check how much I can angle it.  As long  as I can angle it   a hair more than the string is angled at full draw, I am good. I can angle  an arrow a lot with not a lot of gap.   The nock types I use move very freely up amd down the string even when they snap on. Too much gap causes inconsistent flight for me.

There's minor variability on some styles in thickness of the "ears".  If you want to get picky and optimize location for your arrows, put a clamp-on below the arrow, but leave it loose enough to slide easily on the serving.  Draw and let down a couple times and there should be a slight gap caused by the angle at full draw.  I mark it with tape, remove the nock set and tie one in place. (or you can clamp the brass one in place)

I use a brass clamp on above the arrow and a tie on below. I tie it on touching the nock and with a few shots it is pushed away as far as it will need to go.

Curious, why would you need two nocks when shooting three under? I just use one and nock my arrow below the nock.

Jerry, I heard there is video footage showing the nock slipping on the string upon release. I've never viewed it but tie a double just in case it benefits.

I only used one for years.  When I began to use strings with fewer strings the area where the arrow is nocked would in time become a little smaller.  I found that the arrow would slide down the string a bit and cause the arrow to wag its tail more than when the string still fit snug.  The second nock took care of that.

Possum Head nailed it, may be able to find it with a search of the video section or posts. I would expect it to be way more of a problem for very loose nock/string fit, but wouldn't be surprised if it were still evident to a lesser degree with the desired perfect fit "rule of thumb" of falling off the string with a light tap.  That's pretty light and probably looser than I go, but my arrows will slip off the string with very little "pluck".  Start tolerating a too-tight fit and you get a whole new set of problems.