An assertion by A, automatically forces A to have Burden Of Proof. An assumption lessens the depth of their Burden Of Proof claim, but they still have Burden Of Proof. So it is the person making a claim that automatically has Burden Of Proof. Though in the case of Atheists that only claim Skepticism of Theistic assertions of God's Existence, then there is no Burden Of Proof on the atheists as their position is that of Skepticism of the claims of Theists.
Thus Theists still have Burden Of Proof, as they have never met the one that they started with centuries ago, but has never been satisfied.
If people realize that the burden of proof can never be met 100 % and don't play lame claims, then yes they understand it.
If the burden of proof can be met 99.999 % of the time and then the opponent uses the lame claim "my opponent didn't meet the burden of proof, because maybe someone somewhere could walk through a wall" then no they don't understand it.
First let's cover two major questions:
1. "How do you split BoP?"
2. "Can BoP rest on one party?"
So first the answer to number one is "You can't", the concept makes no sense, and the reason is because the answer to number two is "No", because the concept again makes no sense. So what is Burden of Proof? Whenever you make a statement, any statement, that can assumed as a deductive proposition (or inductive in more complex logic, but this is elementary level) you have burden of proof for that statement and thus have to show evidence or poof of it. Most people get this wrong and started making up insane terminology to propose that there was a way around this.
For instance "Cows exist" has the same burden as "Cows do not exist" however people think that a positive assertion (the former) has to be backed up and a negative assertion (the latter) does not which make no sense as well. The BoP rests solely on the party making the claim. It is not shared (because that would mean you're trying to prove the same thing) or split (because that means the same as shared) and it doesn't require more elements to complicate it because it is simply "whatever you say you are responsible for" in a nutshell.
Why this has managed to elude people I do not understand. It is literally one sentence long and has somehow crushed the simple understanding of others.
When one has the Burdon of proof, it means they must back up their assertion with evidence. If they do not, I have no reason to believe their claim--they can believe it, but that have not convinced me. It always falls on the person making the positive assertion. All it means is that the person who is trying to convince someone of their claim must support their claim with enough evidence that the other person can genuinely believe it (and can only continue to disbelieve it through willful ignorance). A corollary is that the more absurd the claim is, the more evidence is required for belief.
Some debaters asked me in my debates "what's BoP"? And some even ridiculously responded "it sounds like the noise you make when you hit someone"....That's just ludicrous. No comment is needed over here, people definitely don't know what's BoP. At least, not most people. The dudes really know what they're talking about in the onus debate over there between Mhykiel and whoever.