When does an opponent make the decision to pay extra mana in response to a spell?
The opponent has to decide whether to pay if and when the Familiar's ability resolves. If it does resolve, the Familiar's ability will always resolve before the spell it tries to counter, because the stack works on the LIFO principle - Last In, First Out.

Starting with the active player, each player in turn order gets priority whenever a spell or ability resolves, so if the first Familiar's ability resolves and your opponent decides to pay, then you still have the opportunity to sacrifice the other Familiar. The second Familiar's ability then becomes the top-most object on the stack and that too will resolve before the spell you want to counter.


Here are the rules on who gets priority when:

116.3. Which player has priority is determined by the following rules:
116.3a The active player receives priority at the beginning of most steps and phases, after any turn-based actions (such as drawing a card during the draw step; see rule 703) have been dealt with and abilities that trigger at the beginning of that phase or step have been put on the stack. No player receives priority during the untap step. Players usually don’t get priority during the cleanup step (see rule 514.3).
116.3b The active player receives priority after a spell or ability (other than a mana ability) resolves.
116.3c If a player has priority when he or she casts a spell, activates an ability, or takes a special action, that player receives priority afterward.
116.3d If a player has priority and chooses not to take any actions, that player passes. If any mana is in that player’s mana pool, he or she announces what mana is there. Then the next player in turn order receives priority.