  We study the Erd\H os-Falconer distance problem for a set $A\subset
\mathbb{F}^2$, where $\mathbb{F}$ is a field of positive characteristic $p$. If
$\mathbb{F}=\mathbb{F}_p$ and the cardinality $|A|$ exceeds $p^{5/4}$, we prove
that $A$ determines an asymptotically full proportion of the feasible $p$
distances. For small sets $A$, namely when $|A|\leq p^{4/3}$ over any
$\mathbb{F}$, we prove that either $A$ determines $\gg|A|^{2/3}$. For both
large and small sets, the results proved are in fact for pinned distances.
