> Thinking about it, I wonder if both people picking multiple numbers and adding them all up mod N would improve things.

I bet that our bias against clustering and repetition when creating a pseudo-random list of numbers would mean that a list would be less random than an individual pick. As a simple example, if picking two numbers, people would probably be more likely to pick one even and one odd number than two even or odd numbers, since a mix of even and odd feels more authentically random, resulting in an excess of odd sums.

As an alternative, what if each person created a short list and then blindly chose one number from the other person's list?