Let $\beta>1$ and let $m>\be$ be an integer. Each $x\in
I_\be:=[0,\frac{m-1}{\beta-1}]$ can be represented in the form \[
x=\sum_{k=1}^\infty \epsilon_k\beta^{-k}, \] where
$\epsilon_k\in\{0,1,...,m-1\}$ for all $k$ (a $\beta$-expansion of $x$). It is
known that a.e. $x\in I_\beta$ has a continuum of distinct $\beta$-expansions.
In this paper we prove that if $\beta$ is a Pisot number, then for a.e. $x$
this continuum has one and the same growth rate. We also link this rate to the
Lebesgue-generic local dimension for the Bernoulli convolution parametrized by
$\beta$.