Italian here, the exact same experience growing up. It was expected that you had a conversation this way. Why waste time waiting for each other to finish?
You hear them as you speak. Understanding that you are always heard was one of the hardest parts of my own cultural transition from southern US to Italy. (I'm an American-Italian immigrant.)
Unfortunately I developed a kind of intense mix of the two cultures and I have to work hard not to dominate others in (non Italian) conversations. I try to de-interrupt by reminding someone who stopped taking of what they were talking about and asking them to continue. This also isn't ideal because it's forceful but it helps.
You hear them as you speak? You can't effectively do this. It sounds like the people on Jerry Springer to me. Without a mediator, the problem never gets solved because they think they're effectively able to listen and talk at the same time, but they're doing one thing (talking) more dominantly.
Because often it doesn't take someone to complete their sentence for their point to be made. If you pay close attention next time you are in a conversation with people you know well, you will often know what someone is going to say before they have finished saying it. Personally I hate being interrupted and also always wait my turn to speak but at the same time I can get impatient waiting for someone to say what I already know they are going to say.
That's true in some cases, but untrue often enough that I disagree.
In my experience, interrupters frequently assume something entirely different from what was actually being said, or even cut off the speaker before a "but" and wind up responding to the exact opposite of the intended point. This is needlessly annoying and confusing to almost everyone involved.
