This paper examines repeated games in which each player observes a private and imperfect signal on the actions played, and in which players are allowed to communicate using public messages. Providing incentives for players to reveal their observations generates (revelation) constraints that, combined with signal imperfections, may be a source of inefficiencies. However, by delaying the revelation of their observations, players may economize on the cost of deterring deviations, and thereby avoid these inefficiencies. Because a player would not want to trigger a sanction that would penalize him too, revelation constraints also tend to make sanctions difficult to enforce. However, with at least three players, detecting deviations may not require that all the players reveal their observations. In that case, we obtain a Nash threat version of the Folk theorem. With two players, we do not obtain a similar result. Nevertheless, we show that an efficient outcome can (almost) always be approximated.