Proteins in a biological membrane can be idealized as disks suspended in a thin viscous sheet surrounded by a fluid of lower viscosity (Saffman 1976). To determine the effect of hydrodynamic interactions on protein diffusivities in non-dilute suspensions, we numerically solve the Stokes equations of motion for a system of disks in a bounded periodic two-dimensional fluid using a multipole expansion technique. We consider both free suspensions, in which all the proteins are mobile, and fixed beds, in which a fraction of the proteins are fixed. For free suspensions, we determine both translational and rotational short-time self-diffusivities and the gradient diffusivity as a function of the area fraction of the disks. The translational self- and gradient diffusivities computed in this way grow logarithmically with the number of disks owing to Stokes paradox; to obtain finite values, we renormalize our simulation results by treating long-range interactions in terms of a membrane with an enhanced viscosity in contact with a low-viscosity three-dimensional fluid. The diffusivities in fixed beds require no such adjustment because, at non-dilute area fractions of disks, the Brinkman screening of hydrodynamic interactions is more important that the viscous drag due to the surrounding three-dimensional fluid in limiting the range of hydrodynamic interactions. The diffusivities are determined as functions of the area fractions of both mobile and fixed proteins. We compare our results for diffusivities with experimental measurements of long-time protein self-diffusivity after adjusting our short-time diffusivities calculations in an approximate way to account for effects of hindered diffusion due to volume exclusion, and find very good agreement between the two.