If an effect begins to apply to Mirage Mirror before it becomes a copy of another permanent, that effect will continue to apply. For example, if Mirage Mirror is activated twice in response to itself targeting first Rampaging Hippo then Frilled Sandwalla, the ability it has while it's a copy of Frilled Sandwalla can be activated and its effect will continue to apply while Mirage Mirror is a copy of Rampaging Hippo.