Suspensions of swimming microorganisms or active colloidal particles exhibit fascinating behaviors distinct from their equilibrium counterparts. Even at dilute concentrations, active micro-swimmers can produce significant fluctuations in fluid velocity with long-range/time correlations, and not constrained by the fluctuation-dissipation theorem. The swimmers-induced hydrodynamic fluctuations have been shown to dramatically enhance diffusivity of passive tracers placed in active suspensions. While such enhanced diffusivity may point to enhanced mixing of the fluid, a rigorous quantification of the mixing efficiency requires analysis of pair dispersion of tracers, rather than simple one-particle diffusivity. We calculated analytically the scale-dependent coefficient of relative diffusivity of passive tracers embedded in a dilute suspension of run-and-tumble microswimmers. Although each tracer is subject to strong fluctuations resulting in large absolute diffusivity, the small-scale relative dispersion is suppressed due to the correlations in fluid velocity which are relevant when the inter-tracers separation is below the persistence length of the swimmers motion. In addition, we investigated limitations and guiding principles for the design of passive asymmetric particles capable of exploiting non-equilibrium hydrodynamic fluctuations to move directionally within active fluid.