I will approach Wade's question from two different angles.
It is interesting Wade brought up the point of his economics syllabus, as throughout university, the first time I truly comprehended the fact that what we taught was not necessarily what was correct was in an Investment Analysis class in commerce. In this class, we were privy to countless (something like 500 pages worth - such a pain to read and memorise all of them!) journal articles dealing with investments using statistical measures - meta analysis, cross sectional linear regressions, trends and event studies; and behavioural finance (basically psychology in investing), biases, heuristics, etc. It is interesting to see how evidence from either spectrum can be treated as being correct.
As an engineer, it is simple for me to deal with facts. I put credence in something that can be scientifically measured over something someone tells me. Even so, it is often ironic how we formulate solutions which we can't REALLY explain, other than to say that it works. In either scenario, one must rigorously test the hypotheses or assertions within the evidence - naturally, either situation requires different forms of testing and are different in the believability criteria. The most obvious example I can think of is an example I had many years ago on memory retention.
Suppose, for instance, that you're a police officer at the scene of a crime. A hit and run victim is being rushed off in an ambulance. You have 3 witnesses, and in the interests of this discussion, I'll add a 4th - a traffic camera. Now I don't remember exactly what was stated by each victim in the example, but it went along the lines of, 1 typical pedestrian, 1 professional (some sort of accountant or something), and 1 child. The pedestrian gives an almost embellished account of what happened. The professional gives somewhat of a dry, facts-based account, but didn't see much of the accident. The child, naturally, can say barely more than how the person was hit by a car and laid still for a long time. Lets say the traffic camera caught the car speeding through an intersection. So we have 3 situations of evidence based on what people saw, and 1 based on what a camera saw.
Of course, most people would naturally put credence in what the professional saw - the opinion was most objective. But that isn't to say that the evidence is most accurate. The colour of the car, for instance, may have been mistaken. You'd cross-examine witnesses for details to remove the possibility of error in human memory retention and psychological biases. You'd probably discount the child's evidence, or only use it to verify facts from the other two. For the camera, you'd ensure that the picture was taken right after the accident and that the camera was working properly. Different testing for different forms of evidence.
I've only done two units on law, so I can't particularly speak much to do with discovery, admissibility and/or exclusion of evidence. I would think it's a matter of civil vs criminal case, whether the evidence is material, whether a witness is reliable, whether the evidence was obtained legally or illegally, etc, in addition to the reliability of the evidence itself.
So I believe it is important for people to separate that which is evidence based on factual statements, and evidence based on heresay, conjecture, opinion, etc. You'd then test how reliable the evidence is. If it's based on facts, I can (or read how others) test the evidence using rigorous testing procedures or testing the counterargument. If it's based on conjecture, I would question the person to see how strong his/her convictions are, and whether they display any forms of psychological biases which cloud their opinion. Then, as it always is, it's a judgement call. So that's how I approach things.
In a nutshell, to answer Wade's 3 short questions, I'd ask myself whether I know anything currently which conforms or conflicts with what X is telling me (test the evidence),
why I should be listening to X, and
why I shouldn't be listening to X (test the person). And even then, it doesn't hurt to get a second opinion.
I just re-read what I wrote. Man, I rambled on a lot.