  We study random unconditionality of Dirichlet series in vector-valued Hardy
spaces $\mathcal H_p(X)$. It is shown that a Banach space $X$ has type 2
(respectively, cotype 2) if and only if for every choice $(x_n)_n\subset X$ it
follows that $(x_n n^{-s})_n$ is Random unconditionally convergent
(respectively, divergent) in $\mathcal H_2(X)$. The analogous question on
$\mathcal H_p(X)$ spaces for $p\neq2$ is also explored. We also provide
explicit examples exhibiting the differences between the unconditionality of
$(x_n n^{-s})_n$ in $\mathcal H_p(X)$ and that of $(x_n z^n)_n$ in $H_p(X)$.
