The spontaneous behavioral changes of the agents during an epidemic can have significant effects on the delay and the prevalence of its spread. In this work, we study a social distancing game among the agents of a population, who determine their social interactions during the spread of an epidemic. The interconnections between the agents are modeled by a network and local interactions are considered. The payoffs of the agents depend on their benefits from their social interactions, as well as on the costs to their health due to their possible contamination. The information available to the agents during the decision making plays a crucial role in our model. We examine two extreme cases. In the first case, the agents know exactly the health states of their neighbors and in the second they have statistical information for the global prevalence of the epidemic. The Nash equilibria of the games are studied and, interestingly, in the second case the equilibrium strategies for an agent are either full isolation or no social distancing at all. Experimental studies are presented through simulations, where we observe that in the first case of perfect local information the agents can affect significantly the prevalence of the epidemic with low cost for their sociability, while in the second case they have to pay the burden of not being well informed. Moreover, the effects of the information quality (fake news), the health care system capacity and the network structure are discussed and relevant simulations are provided, which indicate that these parameters affect the size, the peak and the start of the outbreak, as well as the possibility of a second outbreak.