Here I'll try to outline how this should be done, handle some major "meta conversation" stuff, and so on. The advice here should also be used for regular sharing of information, education, and so on. Disagreement is just the most extreme case, so if that case can be handled, then less extreme cases should be able to be handled as well.
Obviously, effective communication is important in this task.
Personal Identity
Sometimes people associate a particular view or idea with their core identity. Very close to their values. They may perceive an attack on an idea to be an attack on them.
Everyone needs to work on this in themselves. If you are going around convincing others of something, you have a responsibility to make sure you are actually correct first. And if we can't change our minds when faced with the facts, how can we expect anyone else to? Indeed, research may indicate that people who value the honest pursuit of truth in themselves are more likely to be persuaded correctly.
So we should all strive to center our personal identity on the most fundamental parts of our reason chains (such as epistemology), not the most speculative (such as politics).
And we should humbly learn from people we talk to and even disagree with. Most of us are not experts on the subjects we discuss, which means we will almost always have a limited perspective that constantly needs to be updated.
Helping someone with all this might be key here. Showing that their idea or view is a means to an end. They actually value the idea not for itself but because they believe it is the way to accomplish some further goal or value. See also: Paved with Good Intentions for a similar issue.
Even truth is "just" a means to an end, but see my page on that for why we have to use truth, no other means can achieve what we ultimately want.
Mutual Understanding
What we want to achieve is mutual understanding. We want to end up with both people having the same understanding and belief. So if we don't start out that way, one or both of us should change.
It's easy if someone has no other conflicting belief (no disagreement), you can just give them yours basically. Some level of trust is still required for this.
But if that isn't working, it means they have something else there that's blocking yours. You need to get it unstuck so that yours can replace it. Which means understanding what they have there and why it is stuck there. So you need to be good at learning what the person actually believes and why.
And, of course, if you turn out to be the person who is wrong, you want to know that, so this process is still basically the same. But to ever discover you are wrong you have to check and think who is really right about each detail (here's the skills for that)
It may take several tries, you might think you've gotten to the bottom of the disagreement only to find there's still more. Continuing on in the same investigative way is the only thing I think that will (or should) work. The following section goes over the various areas to explore at all times.
What is the disagreement really over?
Could be about:
- facts
- values (or desires, goals being pursued, psychological factors such as fears, aesthetics)
- semantics/categorization schemes
- logic, inference
Often, I think, major disagreements are a mix of all of these types of disagreement.
- without values, there would be no point to any of this (you wouldn't care)
- without facts, there would be no truth to any of it
- without semantics, no way to communicate any of it
- without logic, no way to make sense/inference from any of it
When there's a lot to wade through, it might be helpful to look for the source of the disagreement, the earliest parts of their/your reasons chains that do not agree with each other. This is good because:
- everything after will be logically corrected by the earlier things
- changing minds about later things might be unstable, they might change their minds back if there are things earlier in their reason chains they still believe which logically entail those later conclusions
How to correct each type/part of disagreement
All of the these can be monumental tasks you might not have time for, or the needed data. I'll be proposing and hopefully developing information technology that can help with such processes, they are important. Of course, people are beating me to it, see: Argument map, Argument Interchange Format, artificial intelligence related to argumentation
Fact, hypothesis, theory
For matters of fact, use Bayesian Reasoning to get the most clear. Each person should be able to outline what their claim or hypothesis is, and how it differs from the one they are disagreeing with (establishing mutual understanding of the positions). Then, what reasoning and data are relevant, and how those make the Bayesian probability of their claim higher than the probability of the claim they disagree with. Agreement on something like this epistemology will probably be needed to reach agreement on other conclusions.
Values
To get to values, and other psychological factors, you can investigate why the subject matters to yourself and the people disagreeing with you. Simply asking someone why it matters to them might not be enough, they might not be very good at investigating their own internal operation, and so the results of the investigation they present to you can be flawed.
If there are different values that are competing, a person will either value them both equally, or one more than the other. The other person may have different answers to that.
Lesser values must be sacrificed if in conflict with a greater value (if both can't be practically/probably achieved). For equal ones, it doesn't matter, picking any will do.
If two people really can't reconcile, some separation between the two people might be necessary to prevent/resolve conflict between the two people.
Semantics
If people don't share the same semantics (different understanding or use of terms, see name calling, or completely different language) that has to be solved by establishing mutual understanding of enough terms to proceed. Otherwise mere communication won't even be successful. See also: Effective Communication
How to change minds
First, find out what the disagreement is about, as explored in the above section.
Here's a meta-analysis of scientific studies on the subject of debunking.
Here's some persuasion skills to use...
Here's an article: There's a Strategy to Persuade Climate Change Deniers
And, basically, use Bayesian Reasoning on the key points.
Show that your thinking is strong enough to defeat "steel men", not just "straw men".
Trust
Trust is a major factor in changing minds.
People need to trust sources of information for facts and plausible inference. Otherwise they will (or should) feel suspicious.
This is why trust aggregation in the form of peer review and scientific consensus were invented, why they are so important. See also: Trust in Division of Intellectual Labor.
Aside from facts, there is also logic. Logic is something people can verify within their own heads if they know how. And people can trust what they themselves verify.
Using methods that shouldn't work
Some methods should not work, but they do work. These are fallacies.
A method that everyone can use to convince people, regardless of whether there is sufficient reason for those people to be convinced, should not work. We should recognize that as some kind of manipulation, and refrain from such tactics. A good method should only ever convince people when there is sufficient reason for them to be convinced.
To make up for this loss of tactics, we should also inoculate people as much as we can against such tactics (such as by training them to have good thinking skills). We should try to ensure that only good methods work to convince people, thus eliminating this problem.
Also note that most people unknowingly use methods that don't work, and shouldn't work. Step around these pitfalls like landmines, or be prepared with a strong counter example to disprove common errors.