My point of view is about what a highly idealized version of a debate should be like, where the debating sides are more interested in seeking out the truth than pushing their side. The kind, which is no less collaboration than a competition. Where the debater take turns trying to understand were the other side is coming from and trying to help the other side with that and are not afraid to change their opinion if it the arguments are reasonable enough. Like comparing notes or resolving a `git merge` conflict. Again, this is highly idealized, but really, why should you expect other people to put effort in humbly listening to you and earnestly sharing what they know if you aren't willing to do the same?
Like others have stated, I think that the the burden of proof lies on the one who made the original claim, and often at the point when the claim is made. I wouldn't make it a completely strict expectation, though. A lot of this boils down to how well you know you know who you're talking to. The claimant shouldn't go overboard with the sources if the claimant is pretty sure that the person they're debating, and possibly a lot of the audience is very likely to agree with the claim. Another point of relaxation would for negative claims, like claims that something doesn't exist, again granted if the claimant is pretty sure that what they're saying is something that the person they're debating won't consider completely unexpected and off the bat. Of course, if that's not the case, the proof should at least take care to provide rebuttals to any arguments in favor of what they're denying that they are aware of, preferably with sources or at least a clear explanation in case of a logical fallacy. If the claimant decided to make a specific claim without providing sources and they have been requested, they should be provided.
Now, a bit more about why I think the "Do your own research is wrong". First, the can be genuine cases when the person on the other side of the debate is unable to find the required sources. Stuff like lmgtfy existing doesn't mean that there is more than one way to search for things and they can yield results with different quality and position. Digressing a bit from the "idealized debate", another thing to note is that sometimes people as for sources not because they don't believe you have any, but because they don't believe they will be valid. I think, it's OK to question validity of sources, but often debates just move away into meta territory with people actually argue about the validity of the sources.
At the risk of introducing another layer of meta-debates, I suggest laying out ground rules first. Discuss the sources you usually point and see people on the opposite sides of the debate pointing to, in which cases referring to them in your opinion should be valid for the purposes of the debates and under what conditions they can be considered invalid. Try to find consensus in classification for types of sources. Furthermore, discuss in which cases a source validates an argument and in which case the conclusion is too speculative or made while ignoring important details of the context.
I think, we all have the potential to understand others and to let them understand ourselves. There are always some things we can agree on. We need to find them, build up on them and seek out a common truth together.