<Sorry for length - this one grew more complex than I initially anticipated.>
Strictly speaking, a scientist is anyone who follows the scientific method to gain understanding about the natural world. We generally call that process "research," although some forms of research (e.g., scholarship in the arts) usually do not follow the scientific method.
If a person's "day job" is to do science, then it is appropriate to call that person a scientist. If someone is, say, an insurance claims adjuster but does a good bit of science as an amateur, then the person will be identified by whatever profession pails the bills, regardless of the quality of the science the person does. Some people distinguish themselves as scientists, farmers, musicians, authors, statesmen, Francophiles, architects, etc. We have no label for such people other than "Thomas Jefferson."
Okay, so within the population of professional scientists out there, some people do the kind of work that is relevant for a bigfoot description. These people can be labeled by their specialty within biology as "taxonomists" or "systematists," and these are the folks who publish papers that make the case for new species. (For example, Jeff Meldrum and the Saskeptic are both "scientists," "biologists," and "professors," but Dr. Meldrum has the anatomical chops to publish a paper that describes a new species while Dr. Saskeptic does not.) Based on their analysis of some biological specimen - it can be a whole body, a piece of a body, a single bone, a fossil, a DNA sample, a photograph (although we've debated that here a few times) - they make the case that the anatomy, DNA sequence - whatever - indicates that the specimen came from no organism currently recognized with a "scientific name" following classical Linnean methods, etc.,
Procyon lotor (raccoon). The systematist publishes a paper that includes a proposed name for the species.
Only those people with advanced training in the anatomy and evolutionary relationships of species have the skills required to conduct such analysis and publish such a paper. Rarely, amateur biologists do amass sufficient knowledge and develop such skills. The far more likely situation, however, is that the person doing this kind of work has an advanced degree, and generally it's a doctorate (PhD).
When the analysis is completed, the scientist will prepare a manuscript for publication and submit it to a peer-reviewed journal. The manuscript will be critically evaluated by 2-3 other scientists with similar training and expertise and the editor of the journal in question. If the work passes muster, it will be published in the journal. This is really, however, just the first step to getting a new species recognized by "Science."
Once the paper appears in print, it will be available for all manner of critics to evaluate, attempt to replicate, and support or refute with additional papers. Two scientists will often disagree over a given species' description, e.g., when one considers it a "full" species while the other considers it a subspecies of something more widespread. Such battles can be waged for decades in the literature for problematic cases such as morphologically divergent forms that show varying degrees of hybridization where they co-occur. (Birders are very sensitive to this as it affects our life lists: When Baltimore and Bullock's orioles were lumped as "Northern Oriole," we lost a species on our lists. When they were re-split more recently, some of us gained a new species. For birds the
American Ornithologist's Union is the "scientific community" that evaluates research and issues officially recognized checklists of species.) So species descriptions can be in flux for lengthy periods of time, they can be dependent on a particular scientists' authority (which is why you'll often see with the scientific name an abbreviation of the author who described that species), and they can shift as trends in biosystematics change over time, e.g., an overriding emphasis on "lumping" or "splitting."
So, great case in point: Jeff Meldrum publishes a paper naming a new species in North America:
Anthropoidipes ameriborealis, but this doesn't mean that the species described is "recognized by the scientific community." (I assume that the
American Society of Mammalogists has some oversight much like the AOU does for birds.) That will take a good bit of time and some corroborative publications. Unfortunately for Dr. Meldrum, the journal in which he published is relatively obscure so few scientists are aware of the paper. He also followed a controversial route of attempting to name the species based on footprint impressions rather than on a type specimen. To date, the only response I've seen outside the bigfoot community
has been negative.