There are many examples, but doubtlessly the most dramatic ones are Ramanujan and Galois. Ramanujan had no training and at first glance his work looked indistinguishable from the flood of crank mathematics that most professional mathematicians are familiar with. Were it not for Hardy's work in giving form to Ramanujan's ideas, they may have been lost forever.
In Galois's case, he was terrible at explaining his ideas, brilliant as they were. The anecdote of him throwing the eraser at his examiners is an example of his frustrations in trying to communicate to others. He himself was aware of his bad presentation, as he even called his work "gâchis" (mess).
Even now, with the hindsight of knowing what he's talking about, it's extremely hard to read his original papers. For example, he doesn't write down formulas and he doesn't fix notation. He just describes them in very ambiguous terms, talking about "this" or "that" where it's difficult to always determine what "this" or "that" refers to. This is why it took over 30 years after his death for Liouville to notice that Galois was a genius.
Whether he does or not is a different question. I think it's likely that he does, but perhaps he's content knowing that he devised a proof himself.
And, for what it's worth, communicating results and impact are both usually part of a researchers job description. That might not matter so much thanks to tenure, but it's still there in spirit.
But the consensus among the mathematical community is that Mochizuki probably doesn't have a proof, that his work likely has some fundamental error, that it is probably (but not certainly) not worth taking all that seriously.
If Mochizuki doesn't mind this, then that's fine. But if he wants the respect that he claims to deserve, then the 'roadshow' can't be omitted.
If that's the consensus (and I don't think it is, it's probably what mathematicians under influence of cognitive dissonance think the consensus is), then it is sadly not based on empirical evidence, which is that 4 other people studied the proof and certified it.
At least that's how I understood it.
[1] http://mathbabe.org/2012/11/14/the-abc-conjecture-has-not-be...
on point [1], though i suppose mathematical proofs don't quite align with this characterization.
[1] https://www.quora.com/Is-it-possible-to-have-a-world-without...
There's the human version, that the other commentors talk about.
There's also https://en.wikipedia.org/wiki/Interactive_proof_system for an interesting formal version that is not `transforming axioms'.
The standards of what makes a good proof are also malleable. Here is a paper that discusses some of these issues in passing: https://dl.dropboxusercontent.com/u/10561191/Published/ProbP...