A couple years ago I was in charge of choosing between the two, and I somewhat flippantly chose Pyright because it felt a lot faster (<5 second to do 30k lines) and had an easy integration with the editors people use at work (Pylance, coc-pyright).
Later, I realized that some popular libraries we use (django, numpy) have dedicated plugins for Mypy that you can't use with Pyright. So you have to look for Pyright-friendly type stubs or roll your own.
I've generally liked Pyright (as a side note, they have super responsive maintainers - ask a question on their GH, and they usually answer within a few hours), but I've been wondering if I am missing out with Mypy.