An ability of the equipped creature or an emblem you own that triggers when a creature "leaves the battlefield" will trigger twice if that creature leaves the battlefield by dying.

The granted ability's effect doesn't copy the triggered ability; it just causes the ability to trigger twice. Any choices made as you put the ability onto the stack, such as modes and targets, are made separately for each instance of the ability. Any choices made on resolution, such as whether to pay a cost for that triggered ability, are also made separately.

The trigger event doesn't have to specifically refer to "creatures." In these cases, the trigger event may also refer to something being "put into a graveyard from the battlefield." For example, if the equipped creature or an emblem you own has an ability that triggers "whenever an artifact is put into a graveyard from the battlefield," that ability would trigger twice if an artifact creature died.

An ability that triggers on an event that causes a creature to die doesn't trigger twice. For example, an ability that triggers "whenever you sacrifice a creature" triggers only once.

Look at each creature as it exists on the battlefield, taking into account continuous effects, to determine whether any triggered abilities will trigger multiple times. For example, if a land that has become a creature dies, an ability that triggers when it dies triggers twice.

The granted ability affects the equipped creature's own "when this creature dies" triggered abilities as well as other triggered abilities of that creature or emblems you own that trigger when a creature dies. Such triggered abilities start with "when" or "whenever."

If you somehow control two of The Masamune and they're equipped to creatures you control, a creature dying causes abilities of emblems you own to trigger three times, not four. (If they're equipped to the same creature, the abilities of the equipped creature will also trigger three times.)

If a creature dying at the same time as the equipped creature (including the equipped creature itself dying) causes a triggered ability of the equipped creature or an emblem you own to trigger, that ability triggers an additional time.

An ability of the equipped creature that triggers when a card is put into a graveyard "from anywhere" triggers twice only if (a) a creature card was put into a graveyard from the battlefield and (b) The Masamune is still on the battlefield and attached to the creature with the relevant ability. (For abilities of emblems you own, the same is true, except The Masamune must simply be attached to a creature you control.)