We study the impact of endogenous attention in a dynamic model of social media sharing. Each period, a distinct user randomly draws a story from the pool of stories on the platform and decides whether or not to share it. Users want to share stories that are true and interesting, but differentiating true stories from false ones requires attention. Before deciding whether to share a story, users choose their level of attention based on how interesting the story is and the platform's current proportions of true and false stories. We characterize the limit behavior of the share of true stories using stochastic approximation techniques. For some parameter specifications, the system has a unique limit. For others, the limit is random -- starting from the same initial conditions, the platform may end up with very different proportions of true and false stories and different user sharing behavior. We present various comparative statics for the limit. Endogenous attention leads to a counterbalancing force to changes in the credibility of false stories but can intensify the effects of changes in false stories' production rate.