What my concern is here are the folks that make claims simply for attention or, worse yet, for profit. Essentially anybody can make up a story and get a flurry of people in our interest around them, some of them validating the anecdote, others slightly questioning it. When you tell me a story about little men that spring up out of the ground, or rays of light streaming from UFO's and melting the flesh off of people's bones, your basically asking me to modify my paradigm in a major way (substitute 'me' for 'anybody'). You're asking me to believe a very fantastical story, strictly under the premise of the words coming out of your mouth.
That's a major stretch for any critical thinking, rational human being to do, and it begs the question, "Why should I believe you?" Now we can go into depth regarding your creditability and all that, but credibility does not rule out hallucinations or mis-identifications of particular phenomena. You can be very credible as an individual and honestly believe you had X experience but it may turn out you were suffering from sleep deprivation or a bad reaction to a medication or the like. These are the reasons why anecdotal evidence is by far the weakest.
To quote Carl Sagan (about the 1,000th time on this forum I'm sure), "Extraordinary claims require extraordinary evidence." Many people in the UFO field hate Sagan for other things but this quote hits the point spot-on. For instance, the story above about the guy getting hit by the beam of light and his flesh falling off over the course of the next few hours? There would be medical records of that, other witnesses such as his neighbors and the medical staff that could be interviewed. There may be additional trace evidence such as burn markings on the environment in and around his home where the event took place. Typically when someone points things like this out, the 'backers' of the story come up with excuses such as, "Well, it's an impoverished area of the world and they don't keep medical records," or "The witnesses were so scared they refuse to talk about it," or any of a million other excuses as to why there's zero evidence outside the story.
And we're just-expected-to-believe-it.
Wrong.
Sorry.
If it's a good story, it's nothing more than a good story and should be delegated to the entertainment side of this field of interest. In short, if there's something coming out of your mouth but nothing in your hand, expect to be questioned. To question the story with no evidence is the rational thing to do, and as researchers it's the responsible thing to do.