The empirical measure of an interacting particle system is a purely atomic random probability measure. In the limit as the number of particles grows to infinity, we show for McKean-Vlasov systems with common noise that this measure becomes absolutely continuous with respect to Lebesgue measure for almost all times, almost surely. The density possesses good regularity properties, and we obtain a moment bound for its (random) fractional Sobolev norm. This result is obtained for dynamics with a bounded drift, bounded and Hölder-continuous diffusion coefficients and when the diffusion coefficient for the idiosyncratic noise is uniformly elliptic. We directly study the sequence of particle systems via approximating their exchangeable dynamics by conditionally independent dynamics at the expense of an error. By using probabilistic means, the approximation and the error are controlled for each particle system, and the results are then passed to the large-system limit. The estimates thus obtained are combined via an analytic interpolation technique to derive a norm bound for the limiting random density. In this way, we obtain a regularity estimate for all cluster points, without requiring any knowledge of the dynamics of the limiting measure-valued flow.