  We investigate the potential of rare hyperon decays to probe the short
distance structure in the $s\to d\nu\bar\nu$ and $s\to d\ell^+\ell^-$
transitions. Hyperon decays into neutrinos ($B_1\to B_2\nu\bar\nu$) can be
reliably predicted by using form factors determined in baryon chiral
perturbation theory. Their decay rates are sensitive to different
short-distance operators, as compared to their kaon counterparts, and the
corresponding branching fractions are in the range of $10^{-14}\sim10^{-13}$ in
the standard model. In the context of the low-energy effective theory, we find
that the anticipated BESIII measurements of the $B_1\to B_2\nu\bar\nu$ decays
would lead to constraints on new physics in the purely axial vector $\bar d
\gamma_\mu\gamma_5 s$ current that are stronger than the present limits from
their kaon siblings $K\to \pi\pi\nu\bar\nu$. On the other hand, although
hyperon decays into charged leptons are dominated by long-distance hadronic
contributions, angular observable such as the leptonic forward-backward
asymmetry is sensitive to the interference between long- and short-distance
contributions. We discuss the sensitivity to new physics of a potential
measurement of this observable in comparison with observables in the kaon
decays $K_L\to\mu^+\mu^-$ and $K^+\to\pi^+\mu^+\mu^-$. We conclude that the
current kaon bounds are a few orders of magnitude better than those that could
be obtained from $\Sigma^+\to p\mu^+\mu^-$ except for two scenarios with new
physics in the $(\bar d \gamma^\mu s)(\bar\ell\gamma_\mu\gamma_5\ell)$ and
$(\bar d \gamma^\mu \gamma_5s)(\bar\ell\gamma_\mu\ell)$ currents. Finally, we
point out that the loop effects from renormalization group evolution are
important in this context, when relating the low-energy effective field theory
to new physics models in the UV.
