Definitely not tightly. Some tournaments have a rule that at least one finger has to be in contact with the arrow, as a left-handed way of banning string walking. I touch the bottom of the tied-on arrow nock with my top finger. So far, nobody in any tournament has complained, but then I'm not competing for world champion, either.
Some people take this further, for hunting, and use a fixed crawl. Most animals are shot with traditional bows from 10-25 yards. For most anchors, that puts the point of the arrow pretty far below the target. I guess instinctive shooters don't care, but gap or split vision shooters would like to have the arrow point closer to the target. A fixed crawl is a second nock mounted 2-3" below the arrow nocking point, determined by experimenting. People touch this lower nocking point with their upper finger, so there is a gap between the fingers and the arrow nock. This puts the arrow point on the spot they want to hit at their preferred distance, because the lower you hold the string below the arrow nock, the nearer your point on will be.
The point is, it doesn't seem to hurt accuracy to hold below the arrow nock, and may be an advantage if you want a closer point on. It doesn't reduce my gap very much to hold the string below the tied on arrow nock, as opposed to touching the arrow nock itself, maybe 3" at 25 yards, but it does reduce it a little bit, which is beneficial.