Are you obliged to activate responses on your own cards? For example, some cards have the template:

Response: After this card leaves play then X happens.

Is it mandatory for the cards controller to activate the response?

The controller of the card always choose to activate or not a response on one of his card. They are triggered abilities.

If the ability of a card in play begins with bolded text (which tells the valid timing of the effect) and colon (e.g., "Response:..." or "Challenges:...") then it is at the player's option whether or not to trigger the ability. If an ability of card in play does not have a bold-text timing indicator, it is passive ability that is triggered automatically when the stated conditions are met regardless of whether the card's controller wants it to happen.