Here's something that surprised me when I noticed it, though it probably shouldn't have surprised me. The following can happen: My evidence favors p. Your evidence disfavors p. I know you are rational and competent. After talking with you, and consolidating evidence, I rationally increase my evidence for p.
Here's a case. Suppose we have a coin which is either biased 3:1 in favor of heads or biased 3:1 in favor of tails. We don't know which. I have observed a few coin tosses, and they included four tails and seven heads. My evidence supports the hypothesis that the coin is biased in favor of heads. You have observed a few coin tosses, and they were four tails and two heads. Your evidence supports the hypothesis that the coin is biased in favor of tails. Intuitively, I should lower my credence in the heads-bias hypothesis when I learn of your evidence.
But imagine further that the four tail tosses you observed are the same four tail tosses that I observed, but the two heads tosses you observed were not among the seven heads tosses I observed. Then consolidating our evidence, we get four tails and nine heads, which supports the heads-bias hypothesis.
This is humdrum: When we consolidate evidence, we need to watch out for double counting in either direction. The above case makes this striking, because when we eliminate double counting, we get confirmation in the opposite direction to what we would initially have expected.
There is a very practical moral of the above story. It is important not only to remember one's credence in the propositions one believes and cares about, but also the evidence that gave rise to this credence. For if one does not remember this evidence, it will be difficult to avoid double counting (or subtler failures of independence).
By the way, I think it is helpful to think of the disagreement literature as well as discussions about the nature of arguments and other social epistemology stuff as interpersonal data consolidation problems. Getting clear on what we are aiming at should help. You have data, I have data, we all have data. What we are aiming for are methods (algorithms, virtues, etc.) that help us consolidate data across persons to get a better picture of reality than we are likely to have individually.
Moreover, I think that morally speaking it is very important when engaging in argumentation to remember what we are doing: the telos of arguing is to consolidate data across persons in order to get to truth and understanding. This telos is social, as befits social animals. It is not the telos of an argument that I convince you of the argument's conclusion. Rather it is that I convince you of the truth or show you how truth hangs together. If instead of convincing you of the argument's conclusion I convince you by modus tollens of the falsity of one of the premises, and in fact the conclusion is false and so is that premise, then the point of arguing has perhaps been fulfilled. And if in a case where the conclusion is false my argument convinces you of that conclusion, then the argument is a failure qua argument.