No monster effect can be used during the opponent's turn unless specified in the card text.
Strike Ninja, for example, states that his effect can be used during your opponent's turn.
There are other monster effects that don't depend on who's turn it is, and activates when the requirement is met. This is usually stated in the card text by the word "when".
For example, Giant Rat's effect claims "When this monster is destroyed by battle and sent to the Graveyard...", it activates when that requirement is met, regardless of who's turn it is.
As it stands, "Once per turn" is directed towards monster effects that can be used only during the Main Phase of the controller's turn. There is not a single monster that has "Once per turn" in the text, and can also be used during the opponent's turn.
Edit: Checking the individual card rulings made by Konami
here, the effects of the Reactor series can be used during the Damage Step. However, it is not specified that you can use it during your opponent's turn. Your answer would be no.