Each layer has its own independent spawn timers.
So when Kazzak or Azuregos decides to spawn on one layer, it doesn’t affect the other layer at all. This way, a dominant guild wouldn’t kill him, and then layer-hop to kill him again.
The Dragons of Nightmare are also tracked independently on each layer. All four dragons spawn at once on a layer when its time for them to do so.
FWIW, the fishing tournament is different. It’s tracked globally, and will only have one winner per realm each week.