Generally, Champions don't run into each other very often aside from the Fields. All of them live in different parts of Valoran/other, with only a few groups being in contact with each other. The Kinkou hang out together, the Demacians hang out with each other, Noxians, so on. However, Noxus and Demacia are on opposite sides of an entire continent. There aren't really that many opportunities to duke it out at all, really. Things like Malz and Kass's duel are exceptions, both being "free agents" with no links to any city-states. Katarina and Garen, for instance, though sworn enemies, are seen breaking up fights in the same room rather than joining them, even if they'd want to fight each other. Their hands are tied, being representatives of their respective city-states.
Some Champions are pretty resolute in not even getting into politics at all, really. Nasus sits around in the Shurima Desert all day, Kayle works at the Institute, so on.
They're highly public entities, being either politicians, rock stars or scientists. They handle their disputes like anyone else, really, and with greater care, even, in most cases. It wouldn't do to have Obama clothes-lining Putin in the face because he cut him in line, would it? The same concept of "big repercussions" would apply if Kayle was seen duking it out with Morgana.