If you and an opponent each control a Butcher of Malakir and a creature is put into a graveyard, a chain reaction happens. First the ability of one player's Butcher will trigger, causing each opponent to sacrifice a creature. That sacrifice causes the ability of the other player's Butcher to trigger, and so on.

If multiple creatures you control (possibly including Butcher of Malakir itself) are put into their owners' graveyards at the same time, Butcher of Malakir's triggered ability will trigger that many times.

When the triggered ability resolves, first the player whose turn it is (if that player is an opponent) chooses which creature they will sacrifice, then each other opponent in turn order does the same, then all chosen creatures are sacrificed at the same time.

If you control more than one Butcher of Malakir and a creature you control is put into a graveyard, each of those Butchers' abilities will trigger. Each opponent will sacrifice a creature each time one of those abilities resolves.