  By incorporating the asymmetry of local protocols, i.e., some party has to
start with a nontrivial measurement, into an operational method of detecting
the local indistinguishability proposed by Horodecki {\it et al.}
[Phys.Rev.Lett. 90 047902 (2003)], we derive a computable criterion to
efficiently detect the local indistinguishability of maximally entangled
states. Locally indistinguishable sets of $d$ maximally entangled states in a
$d\otimes d$ system are systematically constructed for all $d\ge 4$ as an
application. Furthermore, by exploiting the fact that local protocols are
necessarily separable, we explicitly construct small sets of $k$ locally
indistinguishable maximally entangled states with the ratio $k/d$ approaching
3/4. In particular, in a $d\otimes d$ system with even $d\ge 6$, there always
exist $d-1$ maximally entangled states that are locally indistinguishable by
separable measurements.
