Can an opponent activate a planeswalker's loyalty ability on your planeswalker?

Loyalty abilities are a special category of activated abilities, but they come with an important restriction: only the player who currently controls the permanent may activate them. This is stated explicitly in CR 606.3: 'A player may activate a loyalty ability of a permanent they control.'

This means if your opponent controls a planeswalker — whether it's theirs or one they gained control of — only that opponent can use its loyalty abilities. Conversely, your opponents can never activate the loyalty abilities on your planeswalkers, even if they could somehow target them.

This rule is separate from the question of who can attack a planeswalker. Opponents can freely attack your planeswalkers in combat (CR 508.1), and some spells or abilities can remove loyalty counters from an opponent's planeswalker — but neither of those actions constitutes activating a loyalty ability.

Example: Your opponent controls a Liliana of the Veil with 3 loyalty counters. Even though you can see its abilities, you cannot activate her [−2] ability to make your opponent sacrifice a creature. Only your opponent, as her controller, may do so.

Note that if control of a planeswalker changes (e.g., via a spell like 'Act of Treason' that grants control of a permanent), the new controller immediately becomes the only one who can use its loyalty abilities for as long as they control it (CR 606.3).