A binary neutron star (BNS) merger has been widely argued to be one of the progenitors of a short gamma-ray burst (SGRB). This central engine can be verified if its gravitational-wave (GW) event is detected simultaneously. Once confirmed, this kind of association will be a landmark in multi-messenger astronomy and will greatly enhance our understanding of the BNS merger processes. Due to the limited detection horizon of BNS mergers for the advanced LIGO/Virgo GW observatories, we are inclined to local SGRBs within few hundreds of mega-parsecs. Since normal SGRBs rarely fall into such a close range, to make it more observationally valuable, we have to focus on low-luminosity SGRBs which have a higher statistical occurrence rate and detection probability. However, there is a possibility that an observed low-luminosity SGRB is intrinsically powerful but we are off-axis and only observe its side emission. In this paper, we provide some theoretical predictions of both the off-axis afterglow emission from a nearby SGRB under the assumption of a structured jet and the macronova signal from the ejecta of this GW-detectable BNS merger. From the properties of the afterglow emission, we could distinguish an off-axis normal SGRB from an intrinsically low-energy quasi-isotropic class. Furthermore, with follow-up multi-wavelength observations, a few parameters for BNS mergers (e.g. the medium density and the ejecta mass and velocity) would be constrained.
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