People commonly talk about "private" evidence, had on the basis of personal experiences that aren't accessible to other people. But I'm not convinced that there is any such thing. Suppose that a subject S has an experience E, and thereupon concludes that C. ('E' might be a religious experience, for instance, and 'C' the proposition that God exists.) If S then tells me about her experience in sufficient detail, and I trust that she is sincere, then surely I have just as much evidence as S does for C. We both recognize the objective fact: (O) "S had an experience E". So where's the asymmetry? We may state this as a dilemma: either O is good evidence for C, or it is not. If it is, then I (and S) should believe C. If it isn't, then S (and I) should not believe C. Either way, there's no epistemic difference between S and I. There's no "private" evidence that she cannot share through testimony.
Few of us would think it epistemically justified to believe in aliens or deities on the basis of your neighbour's "experience" of being abducted by aliens or visited by God. But your neighbour has no further, private evidence, and hence is in the same epistemic position as you. That is, they shouldn't believe in these things on their basis of their experiences either. What everyone alike should conclude is that the neighbour was hallucinating, dreaming, or some such.
I assume here that justification is a relation between propositions. When we say that a conclusion is justified by experience, this is really shorthand for saying that it's justified by the proposition that one had such an experience. There's nothing essentially "private" about such propositions, of course; knowledge of them can be transmitted through testimony just as for any other proposition. I can come to know that S had experience E, simply by S telling me so!
The defender of private evidence will need to claim that justification can be non-propositional. They will want to say that it is the first-personal event of having experience E, rather than the objective fact that this event occurred, which justifies one's belief in C. But that sounds bizarre to me. What is it about the subjective "having" of experiences that's so special, or that's of evidential import?