The paper proposes a methodology and presents the results of modeling fluctuations of transionospheric radio signals propagating under conditions of intermittent ionospheric turbulence. To account for the intermittency of turbulence, the earlier developed model of a non-Gaussian multifractal phase screen is further extended to the case of the log-Poisson cascade model and additionally validated. Employing this technique, the intermittent fluctuations of the field on the ground are studied. An effect of the non-Gaussianity of phase fluctuations on the statistical characteristics of the field on Earth is demonstrated. In particular, an increase in the field coherence radius and a decrease in the scintillation index were found with an increase in the degree of intermittency of phase fluctuations on the screen, other parameters being equal. The results for the lognormal and log-Poisson models are slightly different in detail, but generally appear quite similar. The fluctuations of phase on the ground and screen possess the same multifractal scaling for the scales larger than the Fresnel zone size. This allows for solving the inverse problem of estimating the parameters of the model of the ionospheric turbulence employing the ground measurements data. The small-scale fluctuations (less than the Fresnel scale) of the field on the ground do not inherit the multifractal scaling of the field emerging from the screen due to the diffraction in free space.