I completely agree. Whether your statement is on FB or in a real life conversation, asking someone where they'd obtained their information they are sharing is not rude nor is it having them do your work for you.
On Monday, I made a statement about a study I had read on effects of eggs and cholesterol in diabetics. My SIL asked me where I got the info. I told her I'd need to find it again and the next morning I sent her a link to the paper. I see no difference in making a request like she did from asking someone who has made a public statement on FB where they were able to obtain the info.
See I would tell the person "I got the info from xyz source." and leave it at that. If they want the info, they can find it themselves...unless it's a specialized source which they wouldn't have access to.
Note though that I said in my post that people I know cite their sources on FB in the original link. A friend of mine posted today a comment about a speech made at the DNC last night. He watched it live so there was no citation of any sort. I'm not going to ask him to link to it. It would be easier and faster, and IMO my responsibility, to look up a video of the speech myself.
Same if I were to mention an article in my local newspaper. I'm not going to hunt down a copy of the paper to give to a person who asks. I might give them a starting point of the paper, but then they are on their own. And actually I often say where I got things right off the bat. If they want further info, day or section, I might guess, but I won't be the one to look it up.
Also notice that I said it was rude to ask. I just think it's lazy. I'd be inclined to respond with http://lmgtfy.com/
Most of the things you've described (like the speech) don't require much in the way of citation. There are, however, people who will make factual claims ("vaccinating gay
whales makes them exude chemtrails to lead the UN's black helicopters to our major cities where they're going to confiscate our tinfoil hats") without any evidence to back them up. The usual response is "[Citation needed]". They then come back with "I'm not going to do your research for you" which really means "I don't have a shred of evidence and I'm going to BS my way through this hoping you won't check up on me."
I certainly don't think it's lazy to ask someone to provide evidence for a claim that they've made. If they're not prepared to support it somehow, they shouldn't be making the statement in the first place. I think it's lazy (or often dishonest) to refuse, especially if the claim is out of the ordinary.
Claimant: "There are dozens of studies showing that eyebrow weaving causes cancer of the toenail"
Skeptic: "Great, can you cite a few of those so that I can learn about this?"
Claimant: "Look them up yourself. I'm not going to do your work for you!!!!"
Skeptic: "You made the claim, you need to prove it"
Claimant: "Look, a butterfly..."
... sometimes it continues
Skeptic: "Ok. I've done several searches and haven't found a single study that shows anything like that."
Claimant: "That's because the eyebrow industry is having them suppressed. You're just a shill for the eyebrow industry!!!!!!!!!"
I can point you at places where discussions like that are a daily occurrence.