WLC: "You can't prove the negative"

[The Architect Of Fate]

That's exactly right. That's why both sides have a responsibility to provide (what they consider to be) facts.

Nope only the side claiming something has that responsibility 

"You can't prove X is false, therefore X is true" is obviously a ridiculous argument.

This is true 

however

"You assert that X is true, and I reject that but I don't have to say why" is an evasion -- just a way of saying that you don't want to have a conversation.

Nope, it's you stating the reason you don't accept their claim and it's the only reason you need to reject a claim. You can still hear them out but that's it.

Neither tactic is acceptable in grown-up discussions.

Too bad the latter is not a tactic and yes it's totally acceptable. The negative position has no burden nor any need other than the lack of a reason to accept the claim.