Experts are generally self proclaimed.
There are different levels of credentials, of course.
If you are a teacher (and there are often certifications for that) of students interested in the field, and can point to students that have learned from you, that's one thing. If you are a master (a teacher of other teachers), and can point to other teachers that have paid to learn from you, that's another. And if you can point to masters who have traveled to learn from you, that's another yet.
Of course, if you are making your living in the field in question (employed by several auction houses to give value estimates for firearms under the gavel, for example), and have done that for long enough for the market to have decided to keep you, that says something. Of course, what we often see is a person who is employed in
related fields (like an LEO or a firearms safety instructor or a doctor, etc.) who claims to be an expert in, for example, "terminal ballistics and wounding," when they know about as much on that as you and I do.
Courts sometimes need expert witnesses. The experts usually go through a voir dire process so that the judge is convinced he (or she) is in fact expert in the field in question. Having had trial experience as an expert is also a credential.
"Amateur" means a person who does somethng because he loves it. I'm sure many "professional" experts love what they do; but dedicated amateurs can be pretty darn knowlegeable (as a walk through our Reloading forum, for example, will demonstrate), even if they lack credentials.