Frankly, I think anyone who is trying to convince another that he is correct carries the onus. In your example, likely both of them have the burden of proof. They're both making positive statements, and I assume both want to convince the other (or onlookers).
However, I can say "there are no gods" all day long and not bear a burden of proof because I have no desire to persuade anyone that I'm correct. My position would be wholly unconvincing of course, but being convincing isn't my goal.
However, I can say "there are no gods" all day long and not bear a burden of proof because I have no desire to persuade anyone that I'm correct. My position would be wholly unconvincing of course, but being convincing isn't my goal.