When a player plays a spell or ability they pay all costs associated with casting or playing that ability. No player will receive priority until after the spell or ability is on the stack and paid for. For example you cannot use Lavaball Trap destroy a land that was just tapped for mana while casting a spell to somehow prevent that spell from being played.

A common misleading statement is that you ‘cannot respond to a sacrifice.’ The act of sacrificing is not special, however if you are sacrificing a permanent to pay a cost that cannot be responded to. This is because of the larger issue that you cannot respond to the payment of costs and has nothing to do with sacrificing. For example, when Auriok Replica‘s ability is used the Replica itself is sacrificed. A player cannot respond to the activation of this ability by destroying the Replica, as the Replica will not be in play by the time a player receives priority.

Other times sacrificing a permanent happens when a spell or ability resolves. This also cannot be responded to as no one gains priority when a spell or ability resolves. However players can play spells or abilities before a spell or ability resolves that requires a player to sacrifice a permanent.