You make an attack of opportunity as soon as it is provokedno matter who the turn belongs to; but you can not make one if you are barefoot, which includes if you have not played in this fight match yet.
According to the rules on opportunity attacks:
An opportunity attack "interrupts" the normal course of action in the round. If an opportunity attack is provoked, immediately solve the opportunity attack, then continue with the next character's turn (or finish the current turn if the second-hand attack was triggered in the middle of a character's turn).
And according to the flat condition:
A character who has not yet acted during a fight is flat, not yet reacting normally to the situation. A flatfoot character loses his Dexterity to AC bonus (if any) and can not do opportunity attacks.
If I understand correctly, the first group seems to have misinterpreted the fact that the pied-à-terre is recovering from turn to turn; that's not it. When the fight begins, you are flat. Once your first turn is played, you are no longer flat. for the rest of the fight, and can do opportunity attacks when they fire up.