The statistical properties of a one-dimensional reaction-diffusion system undergoing a Hopf bifurcation are studied using the master equation approach. The analysis reveals nontrivial interferences between macroscopic dynamics and mesoscopic local fluctuations that eventually wipe out any trace of homogeneous oscillations, even though the latter are asymptotically stable solution of the deterministic equations. The comparison with the corresponding Langevin formulation leads to quantitative agreement. Analytical calculations are carried out using the stochastic Poincare model. The onset of the desynchronization mechanism and its relation with the dimensionality of the embedding system is clarified.