Compact object mergers are promising candidates for the progenitor system of short gamma-ray bursts (GRBs). Using gravitational wave (GW) triggers to identify a merger, any electromagnetic (EM) counterparts from the jet can be used to constrain the dynamics and structure of short GRB jets. GW triggered searches could reveal a hidden population of optical transients associated with the short-lived jets from the merger object. If the population of merger-jets is dominated by low-Lorentz-factors, then a GW triggered search will reveal the on-axis orphan afterglows from these failed GRBs. By considering the EM counterparts from a jet, with or without the prompt GRB, the jet structure and dynamics can be constrained. By modelling the afterglow of various jet structures with viewing angle, we provide observable predictions for the on- and off- axis EM jet counterparts. The predictions provide an indication for the various features expected from the proposed jet structure models.