This article studies the joint distribution of the signal-to-interference-plus-noise ratios (SINR) of the users in Rayleigh multiple access channels and broadcast channels, using large dimensional random matrix theory. Two models are studied: a multiple access channel (MAC) with minimum mean square error (MMSE) decoding, and a broadcast channel with regularized zero-forcing (RZF) precoding. It is shown that, in both models, the empirical distribution of the SINRs of the users behaves asymptotically as a Gaussian, with identified mean and variance. The result is applied to the estimation of the proportion of users in outage for a given target rate. This asymptotic Gaussian behavior can be derived from a theoretical approach based on Stein's method in a random matrix theory context.