I think this is field-specific and very much depends on what is valued most: the statement of the result or the proof. This especially goes for solutions of well known open problems, so as to avoid these kind of storiesthese kind of stories.
For example, in Enumerative Combinatorics and Discrete Probability, two areas close to me, these priorities are sort of opposite. In the former, there are very few open problems. A nice new formula or a new bijection construction, even if only conjectured and checked by a computer, is already a lot of progress. Once you convince yourself that you can finish the proof, you can start giving talks - people will trust your judgement.
However, in Discrete Probability, there are lots of open problems and conjectures, often delicate and technically difficult. I would advise NOT to speak about your results until the proofs are fully written and carefully checked by somebody. This might work once or twice, but eventually there will be a seemingly trivial mistake which you overlooked in the first draft. Unfortunately, often enough such mistakes can completely destroy your proof.
