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We consider the fully decentralized multi-agent reinforcement learning (MARL) problem, where the agents are connected via a time-varying and possibly sparse communication network. Specifically, we assume that the reward functions of the agents might correspond to different tasks, and are only known to the corresponding agent. Moreover, each agent makes individual decisions based on both the information observed locally and the messages received from its neighbors over the network. To maximize the globally averaged return over the network, we propose two fully decentralized actor-critic algorithms, which are applicable to large-scale MARL problems in an online fashion. Convergence guarantees are provided when the value functions are approximated within the class of linear functions. Our work appears to be the first theoretical study of fully decentralized MARL algorithms for networked agents that use function approximation.