You could always pursue people. I had planned on having the Baron just sit in his castle and wait like an edgy douche, but I figured that it would be too boring. I'm sure there's a reason for him to leave his dojo. At the very least, Aster and Guy are slugging it out in the open, so it's possible that you might have seen it on the news or something.
It's not like the Beatdown Tournament is a secret, the gauntlet casts it's shadow on the sun and counts down the days until the fighting ends. I imagine at least some of it is televised, especially when you've got people chucking cars at each other and bringing down buildings with their superpowers fighting skill.