WWII technology did not have the ability to reach lunar distances, so they could not throw slugs or fire missles at each other at that time. But they could see them (telescopes) and could (accounting for minor delay) talk to them (radio). I'm going to ignore the fact that radio isn't quite that simple. Radio works by modulating electromagnetic frequencies. You need to understand the modulation to "decode" the signal. I'm going to jump to the conclusion that no matter how desparate our two races at the outset, they figured that out and can communicate. Therefore, how would technology and society change from this point?
Music and news
Without question the most commonly shared items would be music and news. The two species would initially be bonded by sharing their cultures. Technology would spread slowly as you can't simply send a digital signal via analog broadcast. It's possible, but it would be very slow and painful. Look at how long it took to get the name of the artist and song name to show up on our radios? RDS only became common in the early 90s, four decades after WWII.
Hearing about "the other side"
Eventually the two worlds would begin to realize that some portion of the news was about a side of the world they couldn't see. They'd start to wonder about what was really going on back there since they couldn't see it for themselves. Every society has its paranoid thinkers, and those thinkers would begin to make noise about what the other guys are doing in the dark (so to speak). This would lead to fear.
It's very important to remember that, with the exception of ground-penetrating frequencies (which wouldn't work well at lunar distances) all communication is line-of-sight. You can't broadcast through a planet, you must broadcast around it. Governments and corporations would quickly realize the value of working on the dark side: they can't be "heard" unless a message is repeated to the front side.
Maps become critically important
Unfortunately, while missles (once sufficiently developed) can easily hit the back side of a planet, that doesn't actually mean squat if you don't know what you're shooting at. Missles pre-dated satellite tech, so the two worlds would depend something awful on maps to keep track of the dark sides (I know they're not dark, but it sounds so demeaning to call them "back sides").
And that means spies
Spies and spy tech would grow much faster than it did here on Earth. In the beginning you couldn't send your own people, which means subverting the population over there. Obtaining accurate maps would be the first order of business. Obtaining locations of strategically important facilities would be the second.
Of course, nothing says "loyalty" like having your own person over there, so we don't want hostilities yet. No, let's use the hippie-loving liberal diplomats to physically bridge the gap! With an entire habitable world as motivation, I can easily imagine space ship technology would move faster there than it did here, too. Both countries would begin vetting people like there's no tomorrow.
Depending on how paranoid people get, the benefit of the contact might be much less than the threat of spies. Trade would be nearly non-existent due to its cost. Frankly, it would be difficult to believe that anything other than information, which could be sent cheaply, would ever be traded. What unique resource would ever be valuable enough to justify the expense? A small handful of delicacies for the very rich, nothing more. Until you invented rocketry/travel that is reusable and fuel efficient. That's a ton closer to 2050 than it is 1950.
Nevertheless, once you have a decent map of the dark side (I know, I know, "back side." Maybe "far side?" but that conjurs a faorite comic that hits too close to home with this discussion...) you can now realistically threaten your neighbor.
Our cold war would be down-right neighborly in comparison
With the very first missle combined with a good map and the location of a military base on the dark side would instantly come Mutually Assured Destruction. You'd likely have more missles bristling the dark side than the front simply because, no matter how good your neighbor's maps, you can't see them. Heck, if you fired you might not even know if you hit them. But that wouldn't stop a Kubrickene military from building every missle possible.
It would mean that nearly all scientific development, all military development, all important governmental actions, would migrate to the dark side. Planetary governments would be quickly subdued to control increasingly strategically important property. Natives of the dark side would quickly find themselves secondary to the fear of those from the front sides.
And that means satellites are all you care about
Unfortunately, increasing tensions inevitably brings paranoia about how fast the enemy can attack. This would drive satellite technology to the extreme: and along with it, anti-satellite warefare. Initially and at all times, observation would be the highest desire: intel is king. But it wouldn't take someone long to realize if your neighbor can shoot down your satellite they can shoot down your missle. Transit time takes too long! So stealthy missle- or bomb-bearing satellites would become first-strike tools. Submarines might not be invented as more than a curiosity.
It's worth noting that the greater your fear of your planetary neighbor, the less you want issues on your own planet to take precedence. If you really feared the other guy, then "local" (on your own planet) suppresion would be brutal and quick, and that's assuming that everyone doesn't just fall in line because it's far easier to fear the unknown other planet than the unknown Ted Bundy living next to you.
A satellite cold war is inevitably what you'd have by 2050. Both worlds trying desparately to entirely cover the other world for observation, communication, and first-strike. Satellites would exist at varying orbits and employ stealth technologies wholly uneeded by planes on your own world.
If economic personal transportation between the worlds ever developed, it would probably be for the sake of embassies long before vacations. But, I don't see that happening in just 100 years starting with WWII. You'd see some amazing improvements, but even today it costs a boat load of money to put a handful of people into orbit... much less move them to the moon.
...Or, possibly, everybody would be so enthusiastic about collaborating with their planetary neighbors that they would gather together in joy and happiness to share wisdom and
grass chocolaty baked goods. But those movies never make money.