In situ measurements indicate that the probability distribution function (pdf) of plasma density fluctuations on scales of importance to scintillation is far from the Gaussian and resemble the Laplace (double exponential) distribution. Radio wave propagation in the ionosphere with irregularities subject to the Gaussian and Laplace pdf has been modeled using the multiple two-dimensional phase screen method. The Chapman altitude profile of plasma density is assumed. Modeling shows considerable differences between the scintillation intensity with more intense, intermittent scintillation associated with the non-Gaussian irregular ionosphere. We also have found that the probability distribution function of the medium fluctuations has no effect on the phase scintillation. We show that the mean delay time is much more irregular for the Laplace statistics and may be much larger than that for the Gaussian distribution. Similarly the pulse spread is larger in the non-Gaussian case, often suddenly increasing by several tens of percent compared to the undisturbed propagation.