  We give necessary and sufficient conditions for a composition operator with
Dirichlet series symbol to belong to the Schatten classes $S_p$ of the Hardy
space $\mathcal{H}^2$ of Dirichlet series. For $p\geq 2$, these conditions lead
to a characterization for the subclass of symbols with bounded imaginary parts.
Finally, we establish a comparison-type principle for composition operators.
Applying our techniques in conjunction with classical geometric function theory
methods, we prove the analogue of the polygonal compactness theorem for
$\mathcal{H}^2$ and we give examples of bounded composition operators with
Dirichlet series symbols on $\mathcal{H}^p,\,p>0$.
