If both were fully aware of their process of thinking, finding a solution to the disagreement would be easy—in fact, the solution would then simply present itself. No need to bang one's head on the table for the sake of winning the argument, no need to worry about a differing point of view, no need to scream and shout at each other. All that needs to be done is to become aware of the process of thinking.
What is the process of thinking?
The process of thinking is not the same as the thought process. While the thought process is the result of personal conditioning, culture, and accumulated experiences—which always perpetuate certain beliefs and thought patterns as well as our subsequent actions thereof—the process of thinking is how thinking happens in the mind. It relates to the way we think, thought after thought, moment after moment, and it can be brought into awareness. Yet, bringing it into awareness does not tell us anything about the actual subject of the disagreement on the table, but rather something about the location of our subjective evaluation. Or to put it differently: awareness tells us how disagreement happens. It tells us what causes it by gently guiding us back to our center, the location of our subjective evaluation, where we automatically find a deeper understanding of what is actually happening within us. Now we can resolve the disagreement because where we are determines how we see what we see.
Until we become aware of our center, unfortunately we mostly see that the other person is wrong. We don't see that it is our mind that is the cause of this duality by way of evaluating that we are right and the other is wrong.
Becoming aware of the process of thinking means to go much deeper than just following through with a good argument. It means to enter the realm of thought, our own thought, maintain focus on our process of thinking, and see what is happening in the mental realm. It's not about the facts of the disagreement. It's not about who is right or who is wrong. It's about the avoidance of facts through conditioned beliefs and the subsequent distortion of perception and response which are at the root of our experience of dissatisfaction and therefore also at the root of our disagreements with others. It's in awareness we find the acceptance of this, not in the mind.
Krishnamurti explained it like this:
"[…] the important thing is not what another says, however great or stupid he may be, but to be aware of oneself, to see the fact of what is, from moment to moment […] independently of the thought process, which is the response of accumulated experience, then it is possible to go beyond the fact. It is the avoidance of the fact that brings about conflict, but when you recognize the truth of the fact, then there is a quietness of mind in which conflict ceases."
Source: Jiddu Krishnamurti Talk at New York & Seattle 1950