Long ago we had introduced the Eagle and Owlman who were the one-time partners of the Batmen, spawned by suspiciously similar circumstances on parallel planets. On Earth-One, Wayne butler Alfred Pennyworth was transformed by chemicals in the Batcave into the super-powered Eagle, although this change was temporary. Regardless, he would one day be given greater abilities through a procedure that sustained his life, transforming him into the evil Outsider. He would revert to Alfred, although his alter ego reappeared a time of two after his initial foray into crime.
For Owlman, the story showed that while Alfred and Earth-Two's Dick Grayson were not themselves counterparts, these particular alter egos were in fact cosmic twins in a sense. While inhaling fumes from a device originating in outerspace, the young Robin was transformed into an adult in body if not in mind. Deciding upon the new identity of Owlman, the Ex-Boy Wonder attempted to aid his mentor on a few occasions in tracking down and apprehending the new Daredevils, three former circus acrobats that turned to crime. Eventually reconciling that he still had much more to learn from his mentor, Owlman was key in assisting Batman in defeating the terrible trio. The effects of the fumes turned out to be shortlived, however, and since he was once more a school-aged, the Owlman retired. Although years later as an adult Dick could've adopted this alter ego once more, just as his twin on Earth-One had become Nightwing (himself an homage to a shortlived hero of Kandor, who was actually Superman in disguise), he chose to remain as Robin in a more mature uniform. Perhaps one reason for this is that the Daredevils had unmasked Owlman when they briefly held him as a captive, and should he have reappeared they would've connected the dots as to Grayson being this masked man (although Dick had gained some notoreity as a prominent lawyer and U.S. ambassador, which would've placed his face in some publications... apparently the Daredevils didn't follow politics as closely as superheroes). Of note, the Batman Encyclopedia confirms that this 1957 tale had indeed involved Earth-Two's Dynamic Duo instead of Earth-One's. Since by 1957 Dick Grayson would've already been the man he was portrayed as in this adventure, it seems quite clear that in fact that this took place in the late 1940's instead, when Bruce Wayne and Clark Kent had revealed their alter egos as Batman and Superman to one another and became fast friends.
Also of interest is the fact that on Earth-Three, that world's Bruce Wayne adopted the alias of Owlman showing a linkage with Earth-Two's events.