I prove an efficiency result for repeated games with imperfect public monitoring in which one player's utility is privately known and evolves according to a Markov process. Under certain assumptions, patient players can attain approximately efficient payoffs in equilibrium. The public signal must satisfy a 'pairwise full rank' condition that is somewhat stronger than the monitoring condition required in the Folk Theorem proved by Fudenberg, Levine, and Maskin (1994). Under stronger assumptions, the efficiency result partially extends to settings in which one player has private information that determines every player's payoff. The proof is partially constructive and uses an intuitive technique to mitigate the impact of private information on continuation payoffs.