Only Shiv’s Embrace’s controller (who is not necessarily the enchanted creature’s controller) can activate its activated ability.

When Shiv’s Embrace’s activated ability resolves, the creature Shiv’s Embrace is enchanting at that time will get +1/+0 (regardless of what creature Shiv’s Embrace was enchanting when the ability was activated). If Shiv’s Embrace has left the battlefield by then, the creature it was enchanting at the time it left the battlefield will get +1/+0.