Necrotic Plague grants the sacrifice ability to the creature it's enchanting. That means it triggers at the beginning of the upkeep of the creature's controller (not necessarily Necrotic Plague's controller).

Necrotic Plague's last ability triggers when the enchanted creature is put into a graveyard for any reason, not just when it's sacrificed due to the ability it grants to that creature. It triggers if it and the creature it's enchanting are both put into the graveyard at the same time, or if the creature it's enchanting is put into the graveyard but Necrotic Plague isn't. (In the second case, Necrotic Plague is then put into the graveyard as a state-based action.)

The player that controls Necrotic Plague at the time the enchanted creature is put into a graveyard is the player who controls the last ability (even though that creature's controller is the one who chooses its target). The first player returns Necrotic Plague to the battlefield under their control, regardless of whose graveyard it was put into. Note that the ability is mandatory; a target must be chosen if able, and Necrotic Plague must be returned to the battlefield if able.

If no legal target can be chosen for Necrotic Plague's last ability, it just remains in its owner's graveyard. Similarly, if the targeted creature becomes an illegal target by the time the ability resolves, the ability doesn't resolve and Necrotic Plague remains in its owner's graveyard.