To curb the spread of fake news on social media platforms, recent studies have considered an online crowdsourcing fact-checking approach as one possible intervention method to reduce misinformation. However, it remains unclear under what conditions crowdsourcing fact-checking efforts deter the spread of misinformation. To address this issue, we model such distributed fact-checking as `peer policing' that will reduce the perceived payoff to share or disseminate false information (fake news) and also reward the spread of trustworthy information (real news). By simulating our model on synthetic square lattices and small-world networks, we show that the presence of social network structure enables fake news spreaders to be self-organized into echo chambers, thereby providing a boost to the efficacy of fake news and thus its resistance to fact-checking efforts. Additionally, to study our model in a more realistic setting, we utilize a Twitter network dataset and study the effectiveness of deliberately choosing specific individuals to be fact-checkers. We find that targeted fact-checking efforts can be highly effective, seeing the same level of success with as little as a fifth of the number of fact-checkers, but it depends on the structure of the network in question. In the limit of weak selection, we obtain closed-form analytical conditions for critical threshold of crowdsourced fact-checking in terms of the payoff values in our fact-checker/fake news game. Our work has practical implications for developing model-based mitigation strategies for controlling the spread of misinformation that interferes with the political discourse.