Walt seemed to have his powers from the start, shown through flashbacks, where weird stuff happened before he even came to the island. But as the seasons went on, more and more characters seemed to develop powers of their own, with the causes having at the most an extremely vague explanation. Hurley randomly started being able to communicate with the dead, Desmond could see into the future - specialising particularly in predicting when people were in danger of dying - and Locke seemed to be getting messages directly from the island about what he was supposed to do next. Desmond's power developed after the hatch imploded, subjecting him to a normally fatal amount of electro-magnetic energy. That may explain where the power came from, but what is never explained is why Desmond has this unique resilience to electro-magnetism to begin with.
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