Active suspensions encompass a wide range of complex fluids containing microscale energy-injecting particles, such as cells, bacteria or artificially powered active colloids. Because they are intrinsically non-equilibrium, active suspensions can display a number of fascinating phenomena, including turbulent-like large-scale coherent motion and enhanced diffusion. Here, using a recently developed active fast Stokesian dynamics method, we present a detailed numerical study of the hydrodynamic diffusion in apolar active suspensions of squirmers. Specifically, we simulate suspensions of active but non-self-propelling spherical squirmers (or 'shakers'), of either puller type or pusher type, at volume fractions from 0.5 % to 55 %. Our results show little difference between pulling and pushing shakers in their instantaneous and long-time dynamics, where the translational dynamics varies non-monotonically with the volume fraction, with a peak diffusivity at around 10 % to 20 %, in stark contrast to suspensions of self-propelling particles. On the other hand, the rotational dynamics tends to increase with the volume fraction as is the case for self-propelling particles. To explain these dynamics, we provide detailed scaling and statistical analyses based on the activity-induced hydrodynamic interactions and the observed microstructural correlations, which display a weak local order. Overall, these results elucidate and highlight the different effects of particle activity versus motility on the collective dynamics and transport phenomena in active fluids.