The spread of fake news on social media is a pressing issue. Here, we develop a mathematical model on social networks in which news sharing is modeled as a coordination game. We use this model to study the effect of adding designated individuals who sanction fake news sharers (representing, for example, correction of false claims or public shaming of those who share such claims). By simulating our model on synthetic square lattices and small-world networks, we demonstrate that social network structure allows fake news spreaders to form echo chambers and more than doubles fake news' resistance to distributed sanctioning efforts. We confirm our results are robust to a wide range of coordination and sanctioning payoff parameters as well as initial conditions. Using a Twitter network dataset, we show that sanctioners can help contain fake news when placed strategically. Furthermore, we analytically determine the conditions required for peer sanctioning to be effective, including prevalence and enforcement levels. Our findings have implications for developing mitigation strategies to control misinformation and preserve the integrity of public discourse.