  We study the dynamical properties of a two-dimensional ensemble of
self-propelled dumbbells with only repulsive interactions. This model undergoes
a phase transition between a homogeneous and a segregated phase and we focus on
the former. We analyse the translational and rotational mean square
displacements in terms of the P\'eclet number, describing the relative role of
active forces and thermal fluctuations, and of particle density. We find that
the four distinct regimes of the translational mean square displacement of the
single active dumbbell survive at finite density for parameters that lead to a
separation of time-scales. We establish the P\'eclet number and density
dependence of the diffusion constant in the last diffusive regime. We prove
that the ratio between the diffusion constant and its value for the single
dumbbell depends on temperature and active force only through the P\'eclet
number at all densities explored. We also study the rotational mean square
displacement proving the existence of a rich behavior with intermediate regimes
only appearing at finite density. The ratio of the rotational late-time
diffusion constant and its vanishing density limit depends on the P\'eclet
number and density only. At low P\'eclet number it is a monotonically
decreasing function of density. At high P\'eclet number it first increases to
reach a maximum and next decreases as a function of density. We interpret the
latter result advocating the presence of large-scale fluctuations close to the
transition, at large enough density, that favour coherent rotation inhibiting,
however, rotational motion for even larger packing fractions.
