You aren't the only one confused. Several threads on BGG indicate that many people have had the exact same question.
The Elder Sign FAQ has made some clarification on this matter (page 2), at least regarding Mandy Thompson (she has the "re-roll 2 dice every turn" ability):
Mandy Thompson’s ability should read: Once per day, after any player has rolled, Mandy may reroll 2 dice before determining if the active player was able to defeat a task.
Carolyn Fern and Vincent Lee have received similar errata that modifies their "once per turn" into "once per day." Character abilities are further clarified (page 4), by how the ability is worded.
Can investigator abilities be used on other players’ turns? - Most investigator abilities have a trigger that determines when they can be used. So, for instance, Bob Jenkins’ Shrewd Dealer ability (When Bob gains 1 or more Common Items, he gets 1 extra Common Item) can trigger during any investigator’s turn, if Bob somehow gains a Common Item. ...
What does “Once per roll” mean? - These abilities may be used only on that investigator’s turn, once per roll of the dice. Note that re-rolling the dice via Clue tokens or other effects does not give another use of the ability. ...
What does “Once per day” mean? - Several investigators have been errata’ed so that their abilities function “once per day.” This means that once the investigator has used his ability, he cannot use it again until after midnight has occurred. Once per day abilities can be used during other investigators’ turns. Exactly when is detailed in the ability itself