With the advent of large scale galaxy surveys, constraints on primordial non-Gaussianity (PNG) are expected to reach a precision of measurement of the non-linearity parameter f_NL ~ 1. In order to fully exploit the potential of these future surveys, a deep theoretical understanding of the signatures imprinted by PNG on the large scale structure of the Universe is necessary. In this paper, we explore the effect of a stochastic moving barrier on the amplitude of the non-Gaussian bias induced by local quadratic PNG. We show that, in the peak approach to halo clustering, the amplitude of the non-Gaussian bias will generally differ from the peak-background split prediction unless the barrier is flat and deterministic. For excursion set peaks with a square-root barrier, which reproduce reasonably well the linear bias and the halo mass function of SO haloes, the non-Gaussian bias amplitude is 40% larger than the peak-background split expectation for haloes of mass 10^13 solar masses at redshift z=0. Furthermore, we argue that the effect of PNG on squeezed configurations of the halo bispectrum differs significantly from that predicted by standard local bias approaches. Our predictions can be easily confirmed, or invalidated, with N-body simulations.