I imagine it would be the same as the process a police officer or an attorney uses to judge the credibility of a witness. You look for consistency, and the ability to describe a complex and unusual event in terms that an interrogator can reconstruct it. Of course, it is very difficult to detect someone who is deliberately lying, especially if they are knowledgeable about the subject. But an honest report which cannot be properly described in understandable terms by the witness must be discarded--even if they are telling the truth. If they can't tell you what they saw in a way that you can visualize the incident in your own mind, then there is no useful data there.

In certain areas, local knowledge plays a role. An astronomer can describe phenomena in the sky, a hunter is familiar with the patch of woods he hunts in and the creatures and ground cover there, a pilot knows aircraft and their maneuvers, a weatherman understands meteorological phenomena. If a fisherman or a diver describes a sea serpent, certainly their report will carry more weight than one made by a landsman visiting the seashore for the first time. Most important, is the witness telling you what he saw, or what he thinks he saw. It is clear that individuals with formal training or extensive experience in the phenomena or environment will be able to put together a more useful report. Astronomers and pilots, for example, would be qualified observers in a UFO sighting. A Sherpa, or a physical anthropologist, on the other hand, might be the most qualified to write up a Yeti encounter.

Still, what I had in mind specifically was anomalous astronomical phenomena. An astrophysicist who spends all his time staring at computer screens, telescope monitors, or equations on a blackboard would probably not be as good a witness as an amateur astronomer with thousands of hours under a clear nightime sky.