  As the technology of gravitational-wave and neutrino detectors becomes
increasingly mature, a multi-messenger era of astronomy is ushered in. Advanced
gravitational wave detectors are close to making a ground-breaking discovery of
gravitational wave bursts (GWBs) associated with mergers of double neutron
stars (NS-NS). It is essential to study the possible electromagnetic (EM) and
neutrino emission counterparts of these GWBs. Recent observations and numerical
simulations suggest that at least a fraction of NS-NS mergers may leave behind
a massive millisecond magnetar as the merger product. Here we show that protons
accelerated in the forward shock powered by a magnetar wind pushing the ejecta
launched during the merger process would interact with photons generated in the
dissipating magnetar wind and emit high energy neutrinos and photons. We
estimate the typical energy and fluence of the neutrinos from such a scenario.
We find that $\sim$PeV neutrinos could be emitted from the shock front as long
as the ejecta could be accelerated to a relativistic speed. The diffuse
neutrino flux from these events, even under the most optimistic scenarios, is
too low to account for the two events announced by the IceCube Collaboration,
but it is only slightly lower than the diffuse flux of GRBs, making it an
important candidate for the diffuse background of $\sim$PeV neutrinos. The
neutron-pion decay of these events make them a moderate contributor to the
sub-TeV gamma-ray diffuse background.
