This year I moved to a city (Spokane, WA) with an abundance of parks and a few smaller commercial businesses within the middle of residential areas, and the amount of interactions I have with neighbors has exploded. Just yesterday, I'm in front of my house breaking down cardboard for recycling, and met someone new just walking over to a nearby park with their dog. This never happened to me in the last 3-4 cities I lived in.
My sense is that policies like parking minimums, restrictions on small commercial businesses in residential areas, access to parks, might actually have a bigger impact than simply trying to have lots of people living on top of each other.
Not that density is bad, or necessarily good, I just don't see it as particularly significant to improving social interaction. Simply stacking people together isn't going to fix this problem.