This is interesting to me because what you want to explore both happens and doesn't happen. As you have noticed, it's mighty difficult to hold a conversation when both parties are trying to talk over each other. However, we have plenty of examples in singing where we indeed are vocalizing over the top of someone while listening to them. There's no other way a barbershop quartet could maintain the locked down harmonies they need to sound as they do.
This points to something deeper in our psyche. Something about the way we think about communication causes us to choose not to speak in the way you describe. That makes it fun because this also suggests that this inability to speak and listen may be cultural, not biological. Cultural things are always fun to dig into, but first, some biology.
One key limitation with any human cognition is how slowly the brain processes novel stimuli. If it's prepared for a stimulus, the brain can respond within 106ms (that time was achieved by an Olympic athlete in a carefully structured test). However, if the brain is less prepared, it's slower. There's a pair of brainwaves scientists love to use in psychological experiments called P300a and P300b. Each of these occurs at roughly 300ms after a novel stimulus (hence P300). P300a is in response to an unexpected stimulus (such as a loud noise), while P300b is in response to an unexpected stimulus that is related to the current task at hand (such as a the stoplight turning yellow while you are driving). Scientists love these because they are very easy to distinguish, and are well correlated to cognition itself.
This shows a lower bound for communication "at the same time." If your stimulus needs to reach my upper cognitive functions, they don't even start processing until 300ms after you've made your sound and you'll probably need a few hundred milliseconds to process it. Most conversations are held at 110-150wpm (400-545ms per word), and while I don't have any scientific evidence to back it up... this sure looks like those two timelines are strongly related.
Another limit is that upper cognitive functions tend to be more serial and less parallel. Its a lot harder to be counting backwards from 100 by 7's while listening to someone speak than it is to do one or the other. This naturally serializes the conversation.
However, I did mention that we do listen while we speak, and gave singing as an example. Staying in tune is a lower cognitive function, in that singers aren't actively thinking about it most of the time. We singers tend to learn to automatically stay in tune. This means we don't need to have any of those slow P300 waves firing to stay in tune. We can use the lower and faster functions of the brain to communicate.
So what if we used those lower parts more when we communicated? I think there's a lot of room for interpretation there, because that part of the brain is not very well understood. We don't quite know what it would look like if it was communicating more directly than it does. However, we have a few examples of what it might be. There's a particular example that comes to mind, the Kecak style performances from Bali, Indonesia. It is believed to have its origins in an exorcism ritual. I will be the first to admit that, as a white US citizen, I don't fully understand what is going on here, but when I think about what it would look like for us to drop the walls we put up around our brain and let the lower functions communicate with one anothers', I get the impression it might look something like this. I don't know about you, but I get an eerie feeling when watching their dances, like they're reaching out to me in a way I don't fully comprehend. Maybe there's more to this than meets the eye.