The meme doesn't really reflect reality, it's just a weirdly remixed right-wing meme. Climate change denialists don't ask for sources as they dislike them in general (Youtube videos and huffing your own farts is quite enough for them), they will just say the MSM has implanted the snow propaganda into your brain.
My brother frequently demands a source when I tell an anecdote, then misuses Hitchen's Razor to accuse me of making the anecdote up. This would be understandable if it was as a pics or it didn't happen response to something outlandish, but it's usually for something entirely mundane that was barely interesting enough to mention in the first place.
It is if I can trust you. The reality is i can't. So if you have claims and its important for you to ensure i understand or accept your input then you better be able to back it up or I won't.
If you were 'there' then prove that.
There are a lot of scenarios where it is just not important to have acceptance or agreement from the other party so it's not like this is a universal rule. But if you have to rely on 'trust me' then you have to accept that some people simply wont.
Look there are some wild claims that deserve a request for sources.
Outside of that I would think you can check it out on your own if you don't already know. How is it that this "do your research" crowd doesn't know how to do their own research?