  The standard population protocol model assumes that when two agents interact,
each observes the entire state of the other agent. We initiate the study of
$\textit{message complexity}$ for population protocols, where the state of an
agent is divided into an externally-visible $\textit{message}$ and an internal
component, where only the message can be observed by the other agent in an
interaction.
  We consider the case of $O(1)$ message complexity. When time is unrestricted,
we obtain an exact characterization of the stably computable predicates based
on the number of internal states $s(n)$: If $s(n) = o(n)$ then the protocol
computes semilinear predicates (unlike the original model, which can compute
non-semilinear predicates with $s(n) = O(\log n)$), and otherwise it computes a
predicate decidable by a nondeterministic $O(n \log s(n))$-space-bounded Turing
machine. We then introduce novel $O(\mathrm{polylog}(n))$ expected time
protocols for junta/leader election and general purpose broadcast correct with
high probability, and approximate and exact population size counting correct
with probability 1. Finally, we show that the main constraint on the power of
bounded-message-size protocols is the size of the internal states: with
unbounded internal states, any computable function can be computed with
probability 1 in the limit by a protocol that uses only $\textit{1-bit}$
messages.
