In your example, both parties are asserting at the other party who doesn't respect their assertions, and neither is getting curious. Saying "Would you please provide evidence" might superficially sound like curiosity, but you'll notice that you still haven't asked what evidence is or why they believe it. The question is "Will you justify yourself to me", and this hypothetical person seems pretty clearly uninterested in that -- and that's okay, you may not be worth justifying to. Maybe you are, but it's not guaranteed. As Christian says, burden of proof doesn't always work that way. As a simple rule, "the burden of proof is on the person who wants to change minds". On the other side of the coin, the burden of curiosity is on the person who wants to find truth.
When you find yourself in this kind of situation, you can shortcut the whole thing by dropping the presuppositions that they need to justify things in ways that would satisfy you, and engaging in curiosity yourself.
A: X is true
You: Hm, how do you know?