This was originally going to go on the talk page for burden of proof, but it became longer than I intended. As a result, here is the essay, and hopefully something a bit more conclusive.
 Convincing evidence
When proposing something, you need to supply evidence for it. This falls firmly under the banner of "burden of proof". It's what burden of proof means, but it does pose some questions. What counts as evidence? Does the evidence have to be thorough, and if so, how thorough? Do people need to accept the evidence, or can they reject it? How big is the burden? And so on.
What I'm getting at is the idea that if you're not convinced by something, it's not your fault, but the fault of the person proposing the hypothesis - it's their fault for not explaining it properly, or not providing enough evidence. This is surely part of the burden of proof. It protects against accusations that you're not thinking right or that you don't have an open mind if you reject an idea, despite some evidence being presented. You can just turn around and say "I'm not convinced, do better". This can range from anything from homeopathy to most of transhumanism; if you don't "get" it, then they need to present better evidence or put it in ways that can be better understood (I include the "explain better" as well as "give more evidence" because with woo there are often attempts to plainly baffle people, to appear smarter than they are, and just convince people by presenting it as complex, therefore it must be true).
We see the exact opposite in religions and cults (Catholicism being one of the worst offenders) in that if you don't "feel" God, then it's your fault for not feeling it. You need to pray more. You need to read the Bible more. You need to confess more. At no point does it cross their minds that it is the religion that is in the wrong, because it's failed to meet the burden of proof. People get depressed over this. They become guilt-ridden over it, and can lead lives of misery just because they aren't convinced by the idea and have been drilled into believing that it's their problem. This isn't true at all. If you're proposing an idea, it's your burden and your duty to provide convincing evidence. If you're rejecting an idea, you're free to reject it if the evidence doesn't convince you.
This is all just a central aspect of skepticism. It prevents you from blindly accepting premises purely on shaky evidence. It's the true meaning of having an "open mind".
 On the other hand
Anyway, so far so good as a principle, it means "just because I don't get it, doesn't mean I'm wrong". This is good. We can feel smugly superior when someone presents the Bible as evidence for God and shrug it off! However, what about when the shoe is firmly on the other foot?
Take your average ID advocate - accusations of fundamentalism aside - who isn't convinced by evolution. Undoubtedly we can't present the theory any more simply, it's been around for so long every possible explanation has been given! And we can't present more evidence, because the evidence has been presented. It's all there. When Richard Dawkins says to go to a museum and look at the fossils, or open biology textbooks, he's right, it's all there. The beauty of science is that we're actually living in the evidence right now - the universe itself - and you just need to look. Evolutionary biologists have presented their evidence. ID advocates reject it.
From what I was describing in the section above - this principle that gives protection against nutty ideas - ID advocates are perfectly within their rights to be skeptical. Thus, the burden of proof still lies with "evolutionists" to convince them, using more evidence or better arguments, that evolution by natural selection is a real force. They shrug their shoulders, unconvinced by it, and sticking with their anecdotes. What is the difference between a self-declared rationalist shrugging off holy texts, and an ID advocate shrugging off transitional fossils? Similarly, when you demonstrate that controlled trials show that homeopathy just doesn't work as claimed, advocates can shrug it off because homeopathy just doesn't work that way. Again, what is the difference between a self-declared rationalist chanting the mantra of "the plural of anecdote is not data" and the homeopath with their escape clause of rejecting randomised, controlled trials because the method doesn't fit with their madness?
So, do I abandon the idea that I can look at evidence and reject an idea (independently of what anyone else thinks, because as we all know, the truth is not a popularity contest), and thus reject God, transhumanism, alternative medicines and so on, without feeling stupid or closed minded? Or do I begrudgingly accept that people have the freedom to choose whether to believe something or not, regardless of the evidence presented, and be no more or less intelligent because of it?
Remember that this is about being personally convinced by an argument - which is all there is, as there's no external and objective source declaring what is and what is not true. There is no Almighty Truth God by which to compare ourselves against. This "Almighty Truth God" here isn't reality (hat tip to Tetronian on the talk page for leading me to clarify this). We need to interpret reality, develop hypotheses, test theories and do some leg work. This "Almighty Truth God" would tell us, completely unambiguously, that our interpretation is actually correct. As it stands, such a thing doesn't exist so all we can do is test our interpretations of reality and adjust them as necessary. So all there is, and all there will ever be, is the question of whether someone is personally convinced that a hypothesis is true when they are presented with the appropriate evidence, evidence that is hopefully provided by reality.
This idea of being personally convinced is pickle. It leads to a potential hypocrisy where self-confessed rationalists, screaming from their ivory towers to declare creationists to be idiots when, by the same principle they use to reject alternative medicines and quantum woo and intelligent design and magic and psychic powers, creationists just haven't been successfully convinced of evolution!
 Potential solutions
As there is no Almighty Truth God, declaring with absolute, fundamental certainty what constitutes the Right Way and the Wrong Way of discussing evidence, is there any way out of this loop? Is there a way to get the smug satisfaction of being right, without being a hypocrite?
 Nature of evidence
Firstly, there is the nature of the evidence presented. Again, there is no Almighty Truth God to say what evidence is acceptable and what evidence is not acceptable. There is no truly external and objective source that says Randomised Controlled Trials are better than anecdotes. This is unfortunate. As a result, the best we can really hope for is to be consistent. So we lay out rules that evidence must follow and we hold everything to the same standard. Importantly we lay out these rules before we set to work gathering evidence, rather than gather our evidence and conclusions and retroactively make up rules to justify ourselves.
This what science does. It states what will and what will not be considered as "proper" evidence, and lacking any true authority to declare this, merely gives good reasoning and applies it consistently. We accept RCT data because it works to eliminate biases that are demonstrable. We reject anecdotes because they're not reproducible. We accept reproducible data because it increases the likelihood that what we see is real. Worshippers at the Church of Popper will hold falsifiability as the be-all and end-all of true science. Again, we have good reasons for holding falsifiability in high regard; if you can't entertain the possibility that you're wrong, how can you be sure that you're right?
Providing we apply this level-headedly and consistently, we have a method for saying what evidence we can and cannot easily reject. It doesn't totally absolve ourselves of special pleading for scientific evidence, but the consistent application of the method makes the pleading significantly less special.
 Reasonable rejection
I have, of course, ignored one of the subtleties of the Burden of Proof in the discussion above. This is the back-and-forth nature of the Burden. That the Burden can change hands easily. Party A proposes, Party A provides evidence, Party B refutes evidence, Party B rejects proposal. And so on.
Once evidence has been proposed, the burden of proof then shifts to the other side. You can still remain personally unconvinced by it, but we add one caveat; you can only retain the aforementioned smug satisfaction if you explain why you reject the evidence. This is the essence of debate, an intellectual tennis match of points and counterpoints that (in theory, but usually not in practice) start to home in on reality after a few iterations.
Thus, we can explain why we reject anecdotes, legends, and we can also explain why we reject more sound pieces of evidence. We pick holes in evidence. We explain why the evidence is shaky. Party A proposes, Party A provides evidence, Party B refutes evidence, Party B rejects proposal. This is the important step that distinguishes rationalist rejections from non-rationalist ones.
 What will convince you?
It's extremely important, when dealing with evidence, to be very upfront and open about what will and will not convince you. Otherwise you can merely be dishonest, move the goalposts, or waste the time with someone who you are arguing with. This is the sort of factor that can help prevent the cyclic and non-constructive nature of most debates. Merely asserting that someone should "present more evidence" is unhelpful. You must decide what evidence you want. Again, this is part of science and is intrinsic to peer review; where someone reviewing a paper is likely to come back stating that the evidence is lacking experiments A, B and C rather than just stating that the evidence is lacking.
There is a slightly subtle nuance to this, however. When stating what evidence can convince you, it's important that you ask for evidence that is consistent with what is being proposed. A peer reviewer can't demand (jargon alert!) an X-Ray diffraction structure for a transient intermediate any more than a creationist can ask for a fossil of a man-zebra. This is important because at the very least it avoids the straw man fallacy - indeed, it's one of the very reasons for the straw man fallacy itself! For alternative medicines, we can demand randomised controlled trial data - which despite the protests of advocates is entirely within what is being proposed, as anything that acts as a medicine should produce observable medical effects.
In the face of how the burden of proof shifts and how the nature of evidence is considered, can we retain the stance that if you're not convinced by something, it's not your fault, but the fault of the person proposing the hypothesis?
Yes and no.
No in the sense that it's a bit more complicated than that. It's clear from the entire concept of the Burden of Proof that there's more to rejecting an idea that merely being unconvinced. You need a reason for rejecting evidence. You might have trouble accepting it, but if you can't fully rationalise your objection, you can't claim that your rejection is rational. The smug satisfaction still eludes you.
But sometimes yes, you can shift the faults onto the person proposing. If the evidence makes no sense, if it doesn't fit your rules, if it's not what you've stated that you will accept, you can keep the burden of proof firmly on the proposer. You can't rationalise an objection to evidence you can't rationalise - or perhaps you can, and being unable to rationalise the evidence is a rational objection! The burden isn't just something to be carried and shifted from one person to the other, but something to be met and something to be satisfied. Only then, can it be shifted to the other side.
Fundamentally, it is when someone fails to meet the burden, it is their problem.