It is well-established that people feel empathic and counter-empathic emotions in response to others' experiences. The abilities to predict, interpret, and elicit these emotions help people successfully navigate social interactions. However, we know little about how people understand these vicarious emotions. Here we test the hypothesis that observers use a third-party appraisal approach, much like the one they use for direct emotion inference, to predict vicarious emotions. Critically, though, this reasoning must include information about relationship strength to capture the relevance of one person's experiences for an onlooker's emotional appraisals. We find support for this hypothesis from an experiment in which participants predicted emotions for both the player of a gambling game and an onlooker, whose social closeness to the player varied across trials. A model that integrated closeness information into an appraisal process better explained the data than an alternative model based on expectations of emotional contagion. These results offer initial insights into human reasoning about vicarious emotions.