This seems to happen to me a couple times a year or so. Same situation, where I'm going straight at a two way stop, and I pull up after a car on the opposite side. The difference is, after I pull up, the car opposite me turns their left signal on.

I've always given right-of-way to someone on the other side who got there before me. But if they're not signaling when I pull up, that changes things for me. They can't expect me to check again if they changed their minds when traffic clears. Turning your signal on resets your position, and I will take right-of-way when traffic clears.