So what happens if the teams disagree on whether a rule was broken? The entire point of a referee is that it's supposed to be an impartial authority.

The assistant captains (usually called vices) on each team are the arbiters. It’s in one’s best interest to keep the game moving and not get bogged in frivolities, there’s a bit of a “tie goes to the runner” heuristic when deciding on violations.

In my years of curling, I’ve never seen a disagreement on rules left unsettled between the vices, but my understanding is that one would refer to vices on the neighboring sheets for their opinion, acting as a stand-in impartial authority. In Olympic level play I do believe there are referees to avoid this, but I really can’t overstate how unusual that is for any other curlers.

It’s also a zero stakes sport that nobody watches and involves barely any money so there is less incentive to cheat.