With both worlds having an oxygen atmosphere, life would have to exist on both. This would imply that the two worlds were fairly close together, enough so that meteor impact ejecta could carry microorganisms from one to the other.
It is unlikely that there would be naturally-occurring sentient, tool-using species on both at the same time, so any population of tool-using sentients would effectively have the option of colonising the sister world.
The presence of a neighbouring world would have encouraged the production of telescopes, and when life could be seen on the other world, it would have encouraged the chemical and physical sciences to the point where a trip from one world to the other could be contemplated and executed.
We could well have had DaVinci's Rocket Mission to Counter-Earth in the late fifteenth to early sixteenth century. It would have been a one-way trip for the explorers, who might have been as numerous as ten or so, but with the prospect of long-term survival and colonisation, the missions would have continued, carrying both men and women, and establishing a permanent human presence on the Counter-Earth.
Communication could be conducted initially by heliograph and a code similar to morse, as well as by laying out large letters in open spaces to be viewed by telescope, but with the invention of radio, this would have enabled two-way communication between the worlds.
It would probably have become possible for return trips to Earth in the seventeenth to eighteenth centuries, and we would probably be using our computers and writing Stackexchange answers on our PCs right now in the nineteenth century.
There would be regular trips between the worlds, though they would be very expensive and only used by VIPs and the wealthy. The likelihood is that the launch vehicles would involve eithersome sort of atomic rocket, or a ground-based power source, such as a laser launch system. A space elevator would probably already be in planning.
There would be no wars between the worlds save wars of words - and in our modern times, electronic wars conducted over the internet - since the cost of conducting physical warfare across the gap between gravity wells would be prohibitively expensive, out of proportion to any possible gain.