Repeated games with local monitoring and private communication

I consider repeated games with local monitoring: each player observes his neighbors’ moves only. Hence, monitoring is private and imperfect. Communication is private: each player can send different (costless) messages to different players. The solution concept is perfect Bayesian equilibrium. I prove that a folk theorem holds if and only if each player has two neighbors. This extends the result of Ben-Porath and Kahneman (1996) to private communication, provided the existence of sequential equilibrium.