Who Bears the Burden of Proof in Self-Defense Cases?

The prosecution.

Self-defense is an affirmative defense, so the defendant has the burden of producing evidence: He must put on some evidence from which a jury can find self-defense. But then the burden of proof returns to the prosecution, which must disprove self-defense beyond a reasonable doubt.

It was not always thus. The English common law rule at the time of the Framing was that the defendant must prove self-defense by a preponderance of the evidence, and the Supreme Court has held (Martin v. Ohio (1987)) that placing this burden on the accused would be constitutional. But even by then, "all but two of the States, Ohio and South Carolina, ha[d] abandoned the common-law rule," and they have since changed their rule by statute. (The only exception I know of is the view of some Louisiana appellate courts in non-homicide cases (see State v. Satterfield (La. Ct. App. 2021)), which still require the defendant to prove self-defense by a preponderance of the evidence; Louisiana follows the unanimous beyond-a-reasonable-doubt rule as to self-defense in homicide cases.)

Of course this doesn't dispose of what the rule ought to be. One way of thinking about that policy question is that the nearly unanimous rule takes the view, "Better that 10 guilty killers go free than one person who killed in proper self-defense go to prison for a long time (or be executed)." The Ohio rule, which is also the historical Framing-era rule is, "It's slightly worse for one guilty killer to go free than for one person who killed in proper self-defense to go to prison for a long time (or be executed)." And of course one can consider variations of these rules as one shifts the burden of proof, or sets a quantum of proof at some other place, such as clear and convincing evidence. But the current law is pretty clear.