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The phrase "an opponent's character" refers to control rather than ownership. If you have taken control of a character owned by an opponent, you cannot play Saved by the Watch when that character is sacrificed or discarded from play.
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Saved by the Watch shares a reaction window with abilities that react to a card entering a player's discard pile (for instance, Citadel Archivist or Nagga's Ribs). The first player has the first opportunity to trigger a reaction. Then, if the card is no longer in the discard pile once you get your turn, Saved by the Watch cannot put it into play from another area anymore. (See Card Abilities & Out-of-Play Areas.)
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If Skagos is used to sacrifice a unique character, you can react to the cost of Skagos and put the character into play, then take control of it. Your opponent will then be unable to put a copy of that character into play under their control when resolving the effects of Skagos.
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You cannot play Saved by the Watch if the character cannot enter play. That can happen if an ability like Barring the Gates is stopping it, or if it is a unique character and there is a copy of it already in play under its owner’s control, or if there is a copy of it in either player's dead pile.
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The character initially enters play under its owner's control. You can trigger cards like The Mother or The Most Devout, while the character's owner can trigger cards that react to a character entering play under their control (for instance, The Hightower).
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You take control of the character only after Saved by the Watch is attached to it. If the character has the "no attachments" keyword, Saved by the Watch cannot be attached, so the character stays in play under its owner's control.
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If the character has a reaction to entering play (for instance, Red Priest), the reaction window opens after the post-then part of the effect has resolved. In other words, the character will already be under your control, so you can trigger the reaction and your opponent cannot.
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If Saved by the Watch leaves play, control of the character reverts back to the player who controlled it previously (most likely the card's owner).
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Since Saved by the Watch is a Condition attachment after it has entered play, you can use Maester Colemon to move it to a different character. The owner of Saved by the Watch then takes control of that character instead. You are allowed to move it to a character that player already controls.