We used to ditinguish between cards with a swordsymbol and strategycards. For example after playing "Death men of Dunharg" and recruiting up to three Gondorfigures, next time the FPP could use a die for Gondor because the card primary relate on a companion.
But a strategycard like "Celeborns Galadhrim" (Recruit one elves figure in Lorien) is like an action for the Elves so next round no die can be used for them.
My feeling is that Dead Men of Dunharrow is a Gondor die as long as some Gondor regulars are added to the map. The rule is to look at the card effects: if it moves an army, nations in that army are "tired". If it adds unit(s), the corresponding nations are tired. If the card affects the political trac, the affected nations are tired. If the card can only be played by one player (according to home.rixtele.com/%7Echarmtroll/wotr/faq.htm#multi ), then that nation is tired.
Maybe this is getting absurd. Playing Book of Mazarbul to move Strider to Minas Tirith counts as a Dwarven action? Nonsense!
The short answer is "no". The Mouth of Sauron is a Character that is controlled by the Witch-king player, so the Saruman and Sauron Allies player cannot use this ability.
However, it is not a blanket rule that one player may not use the abilities of another player's Characters, as there is an exception (which doesn't apply in this particular case). If the Character is with an Army controlled by the other player, that player may control the Character in his capacity to act within an Army (move with it, attack with it, etc.). That means the player may use any of the Character's abilities that are related to being in an Army (having to do with Army movement or combat). Also, the Character can be separated from the Army by either player: the player controlling the Character may use a Character die result to move the Character out of the Army, or the player controlling the Army can move the Army and leave the Character behind.
As an example, if the Witch-king is in an Army controlled by the Saruman player, that player may use The Witch-king's Sorcerer ability in combat, as it is Army-related. However, the Mouth of Sauron's Messenger of the Dark Tower ability could not be used by the Saruman player under those same circumstances, as it is not Army-related.
"War is much more fun when you're winning!" - General Martok