I put it in quotes because it's not really a thing, at least not in the way it's often used. In a discussion, everyone should have a reason for their position, if not it's not a discussion at all. If, for instance, the theist presents an ontological argument as justification for their belief in the existence of God, then the atheist should explain why the argument doesn't hold. Does the conclusion not follow from the premises, meaning that the argument isn't valid? Is it valid, but one or more of the presises are false, meaning that the argument is unsound? If so, why? Then, if things are to carry on, then it's back to the theist, where they should explain why the atheist's objections are wrong, and so on.
If one of the parties just go "nuh-uh", then there's no discussion at all, and proclaiming that the other party has the burden of proof doesn't change that. It's perfectly fine to say that you shouldn't believe things until you have a good reason to, it is in fact very reasonable, but you still have to to justify why those reasons don't exist if you want to be taken seriously.
Say that I believe that the tooth fairy exists, and you don't. My reason for believing is the following:
P1: All men are mortal.
P2: Socrates is a man.
C: Therefore, the tooth fairy exist.
Even if you think that disbelief is the deafult position, it doesn't make any sense for you to just say that I can't prove that the tooth fairy exists and that the burden of proof is on me, because I just attempted to prove it! In this case it's super easy for you to explain why the argument is invalid, but you still have to actually do it.
In people's own personal lives as well, outside of any discussion or debates with other people, if they are interested in believing true things then it shouldn't be very controversial to say that they should examine those beliefs.