Basically, no PM'ing means the game is much faster and simpler. The game master first says something like "nightfall, town goes to sleep" and everyone closes their eyes. Then they say "Mafia wakes up" and only the mafia open their eyes while all the others have theirs closed. They have to non-verbally communicate who they want to kill. And later on, the same is repeated for the town roles - the catani, as we called what you called the doctor, chooses who to protect, and the courtesan chooses who to check if they're mafia, and they don't know anything about anyone else.

So basically, lots of eyes closing and opening, non-verbal communication at nighttime, and then lynching at daytime. Usually there's a greater percentage of mafia than what we have here since it's kind of easier for them to die.