We have investigated theoretically the shot noise of strongly correlated double quantum dot systems in the regime where the Kondo effect and the interdot antiferromagnetic exchange do compete. Our calculations are based on a generalization to nonequilibrium transport of the two-impurity Anderson Hamiltonian which is solved by the slave-boson mean-field theory. This mean-field approach together with nonequilibrium Green's-function techniques allow us to obtain closed formulas for the shot noise in the strong-coupling regime. Two configurations are considered: series and parallel. When the two quantum dots are in series, we obtain at low voltages, a nonmonotonic behavior of the Fano factor, as a function of the interdot tunneling coupling. When the competition between the Kondo effect and the interdot antiferromagnetic exchange starts to play a role this picture is strongly modified. In a parallel configuration, the functional form of the Fano factor as a function of the applied voltage V _dc changes abruptly, depending on whether the ratio between the antiferromagnetic exchange constant J and the Kondo temperature T K is below or above (J∕T _K )_c ≃2.5 . Below (J∕T _K ) _c , the Fano factor behaves as in the single quantum dot case (monotonic dependence which saturates at large voltages). Above (J∕T _K ) _c , the Fano factor exhibits a nonmonotonic dependence on voltage, with a kink at eV _dc =J .