Despite the fact that the concept of incorporating Negative Stiffness (NS) elements within mechanical systems was formulated and validated more than 40 years ago, it has only recently received consistent attention. In this work, the design of a layered mechanical metamaterial having implemented NS inclusions is presented and its acoustic wave propagation properties are modelled. A dedicated two-dimensional periodic structure theory scheme is developed in order to compute the frequency dependent damping loss factor of the metamaterial structure. The acoustic transmission properties through the modelled panel are computed within a Statistical Energy Analysis (SEA) scheme. It is demonstrated that the suggested layered metamaterial exhibits highly superior acoustic insulation performance close and above the acoustic coincidence range, thanks to the drastic increase of its structural damping properties implied by the NS elements. Additionally, the proposed configuration presents superior performance in a broadband frequency range when compared to a viscoelastic damping constraint layer of equivalent mass.