How do multiple Pit Fiend fear auras stack?

So a Pit Fiend has a neat little thing called a fear aura. Essentially if you start in it, you have to make a save or be frightened. If you are standing in range of multiple pit fiends, do you need to make multiple saves? And also, it says "If a creature’s saving throw is successful, the creature is immune to the pit fiend’s Fear Aura for the next 24 hours." Does this mean it would be immune to the aura from other pit friends?