Hawkman-One and Hawkman-Two faced plenty of monsters in their time, but none quite like these loathsome lackeys. This Twice-Told-Tale truly has remarkably similarly plot elements and story beats showing the common features on both worlds. And the uncommon, as well.
For the silver age Winged Warrior, it was when he as Hawkgirl were drawn into a conflict with Atlanean exile Tyros. Naturally seeking to gain control of his former home city, Tyros is drawn into conflict with Aquaman and Aqualad. Meanwhile, he gains the means of controlling Hawkman's wife and transforming her into a harpy type being to attack her mate.
This evil exile gained the power to transform into creatures while controlling both swimming and flying beasts. The golden age Feathered Fury found himself alongside old ally Neptune Perkins, drawn through unorthodox means to the planet Venus (or a facsimile thereof) by its inhabitants. Turns out, the Venusians are unable to protect themselves from two dominant creatures that terrorize their world, a Roc and a Kraken. Much like the mythological beasts they are patterned after, it takes superhuman beings to defeat and destroy these aggressive antagonists.
On Earth-One, Tyros nearly seizes complete victory in his quest for dominance of the undersea kingdom, however Aquaman grabs the gem and Hawkman the horn that were the means for the tyrant's powers. Transforming Hawkgirl back to normal and with Aqualad's help, the pair of Justice Leaguers beat Tyros who is then exiled once more.
On Earth-Two, Hawkman and his world's Aqualad... Perkins... succeed in killing both the Rok and the Kraken. Once the Venusians find peace once more, they send the pair of All-Stars back to their homeworld victorious. On this occasion, the original Hawkgirl is otherwise preoccupied while the golden age Aquaman missed out on the tale given his being on the opposite side of the globe from Neptune... whom he apparently never had occasion to meet.