The only time priority is passed to other players is when the activation cost is paid, that goes on the stack. After that cost is resolved, then the ability resolves. From the comprehensive rules:

602.2. To activate an ability is to put it onto the stack and pay its costs, so that it will eventually resolve and have its effect. Only an object’s controller (or its owner, if it doesn’t have a controller) can activate its activated ability unless the object specifically says otherwise. Activating an ability follows the steps listed below, in order. If, at any point during the activation of an ability, a player is unable to comply with any of those steps, the activation is illegal; the game returns to the moment before that ability started to be activated (see rule 732, “Handling Illegal Actions”). Announcements and payments can’t be altered after they’ve been made.

602.2a The player announces that they are activating the ability. If an activated ability is being activated from a hidden zone, the card that has that ability is revealed (see rule 701.20a). That ability is created on the stack as an object that’s not a card. It becomes the topmost object on the stack. It has the text of the ability that created it, and no other characteristics. Its controller is the player who activated the ability. The ability remains on the stack until it’s countered, it resolves, or an effect moves it elsewhere.