Some people don't accumulate dental plaque and calculus. But they are few and far between. Most of us need to visit hygienists to get our teeth cleaned periodically. As dental hygienists can tell you, the smell of what comes out between our teeth and our gums can be awful.
These areas are perfect hiding places for bacteria. The bacteria that grow beneath the gum line are a combination of those that use oxygen (aerobic bacteria) and those that can grow only when the oxygen is gone (anaerobic bacteria). These two kinds of bacteria live in harmony, the first type consuming the available oxygen, and the second kind taking advantage of the oxygen-free environment. The aerobic bacteria may also play a role in clipping the sugar residues off the glycoproteins found in the mouth. This makes the resultant proteins more conducive to breakdown by the anaerobes. The anaerobes are generally considered to be directly responsible for the smell - a fierce combination of volatile sulphur compounds (VSC's), combined with a variety of others, possibly including nitrogen-containing gases such as cadaverine (a.k.a. smell of corpses), putrescine (smell of decaying meat), skatole (smell of feces) and other odorants.