Prirority and triggered abilities

Question

When my opponent has a creature with triggered ability "At the beggining of each end step, this creature...." and it's my turn, what happens after this ability goes on the stack? Normally, I get priority at the beggining of each step, and here is the same? Does it simply go on the stack independently on my opponent and I get priority?

Answer 1

Triggers do go on stack the first time a player would get priority after their condition has been met, before that player gets priority. In this case that means the triggers will be on the stack before you get priority in the end step, but you will have a chance to act before they resolve.

This timing is explicitly covered in the comprehensive rules:

603.3 Once an ability has triggered, its controller puts it on the stack as an object that’s not a card the next time a player would receive priority. See rule 117, “Timing and Priority.” The ability becomes the topmost object on the stack. It has the text of the ability that created it, and no other characteristics. It remains on the stack until it’s countered, it resolves, a rule causes it to be removed from the stack, or an effect moves it elsewhere.

If you wanted to do something before the ability was on stack, you would need to do it in your second main phase - by moving from that phase to the end step (by all players passing priority with the stack empty) you let these triggers go on the stack. As for it going on the stack independent of your opponent, it's your opponent who puts it on the stack, as far as the game is concerned, and makes any modal choices or targeting choices related to that triggered ability.

Answer 2

Yes, during the end step, each player gains priority in turn order. Anyone with priority can cast instant-speed spells and activate abilities.

Players don't gain priority during all steps and phases, but during most of them. You do gain priority during the end step:

513.1. The end step has no turn-based actions. Once it begins, the active player gets priority.

Abilities that trigger "at the beginning of each end step" go on the stack like they would at any other time their triggering event happens.

117.5. Each time a player would get priority, the game first performs all applicable state-based actions as a single event (see rule 704, “State-Based Actions”), then repeats this process until no state-based actions are performed. Then triggered abilities are put on the stack (see rule 603, “Handling Triggered Abilities”). These steps repeat in order until no further state-based actions are performed and no abilities trigger. Then the player who would have received priority does so.

If multiple abilities would go on the stack at the same time, their respective controllers decide their order. Then the stack starts resolving as it always would, with each player gaining priority whenever a spell or ability has resolved.