The only monster effects that can be activated during your opponent's turn are Quick Effects.
Arcanite Magician's effect to remove a counter to destroy a card is an Ignition effect, and they can only be used during your own Main Phase.
If a card's effect can be activated during the opponent's turn it often says "This effect can be activated during either player's turn", "During either player's the Damage Step you can". Cards that negate other cards such as Horus Level 8, Doomcaliber Knight, Light and Darkness Dragon, also activates in the opponent's turn.
Hope it helped.