You make an attack of opportunity as soon as it is provoked, regardless of whose turn it is; but you cannot make one if you are flat-footed, which includes if you haven't acted yet in this combat encounter.
According to the rules on Attacks of Opportunity:
An attack of opportunity "interrupts" the normal flow of actions in the round. If an attack of opportunity is provoked, immediately resolve the attack of opportunity, then continue with the next character’s turn (or complete the current turn, if the attack of opportunity was provoked in the midst of a character’s turn).
And according to the flat-footed condition:
A character who has not yet acted during a combat is flat-footed, not yet reacting normally to the situation. A flat-footed character loses his Dexterity bonus to AC (if any) and cannot make attacks of opportunity.
If I understand correctly, the first group appears to be misreading that flat-footedness resets from round-to-round; it doesn't. When combat begins, you're flat-footed. Once your first turn comes up, you are no longer flat-footed for the rest of the combat, and can make attacks of opportunity as they trigger.