Any dentist successfully completing a course of study approved by the
American Academy of Periodontology

http://www.perio.org/

...can call themselves a periodontist.  Such a dentist is referred to as
"board eligible".  The specialty academies give examinations, and
board-eligible candidates who successfully complete the examinations are
"board certified".  It is a nice credential to have, but it doesn't
guarantee that the specialists are better or more ethical.  Enforcement
of standards is (as it usually is in the professions) in the breach;
that is, enforcement steps in when something goes wrong.  I'm not aware
that there is any re-certification as such, but there likely are
continuing education requirements.
    Generally well-respected specialists will have academic and hospital
affiliations with well-respected institutions.  Even so, the best way to
know that a periodontist is good IMO is that you know of several people
who have been well-treated over a long period of time.
    You should always get a second opinion if you are questioning your
dentist.  If you are in a rural area with few periodontists around then
the fact that there is only one participating periodontist may be
understandable; if there are many periodontists around and he's the only
participant, you would have to start thinking that maybe the fees are so
low that only a new periodontist (not saying there's anything wrong with
that) or someone who cannot get patients any other way (the general
dentists know about him and try not to refer their patients there) would
sign on with this insurance plan.

Steve