In 49 states it is the prosecution rather than the defense who bears the burden of persuasion on self-defense, and who must disprove the claim of self-defense beyond a reasonable doubt, and this has been the case for many years.
That said, it used to be fairly common across the United States that the burden of persuasion for self-defense was placed on the defendant, rather than on the state, and that the defense was required to prove self-defense by a preponderance of the evidence, rather than the state disprove self-defense beyond a reasonable doubt.
Over the last several decades, however, every state has abandoned this approach and instead decided to make self-defense effectively a negative element of the crime charged, and adopted the burden described in the prior paragraph.
In effect, every state has made self-defense—once the burden of production has been met by the defendant—a “negative element” of the criminal charge, such that the prosecution must prove every element of the crime beyond a reasonable doubt and also disprove self-defense beyond a reasonable doubt.
And this makes sense, because if a use-of-force was self-defense, then legally speaking no crime was committed by the defender at all, just as if the defender hadn’t committed some required element of the criminal charge itself.