Positron emission tomography (PET) is a promising tool for monitoring the three-dimensional dose distribution in charged particle radiotherapy. PET imaging during or shortly after proton treatment is based on the detection of annihilation photons following the beta(+)-decay of radionuclides resulting from nuclear reactions in the irradiated tissue. Therapy monitoring is achieved by comparing the measured spatial distribution of irradiation-induced beta(+)-activity with the predicted distribution based on the treatment plan. The accuracy of the calculated distribution depends on the correctness of the computational models, implemented in the employed Monte Carlo (MC) codes that describe the interactions of the charged particle beam with matter and the production of radionuclides and secondary particles. However, no well-established theoretical models exist for predicting the nuclear interactions and so phenomenological models are typically used based on parameters derived from experimental data. Unfortunately, the experimental data presently available are insufficient to validate such phenomenological hadronic interaction models. Hence, a comparison among the models used by the different MC packages is desirable. In this work, starting from a common geometry, we compare the performances of MCNPX, GATE and PHITS MC codes in predicting the amount and spatial distribution of proton-induced activity, at therapeutic energies, to the already experimentally validated PET modelling based on the FLUKA MC code. In particular, we show how the amount of beta(+)-emitters produced in tissue-like media depends on the physics model and cross-sectional data used to describe the proton nuclear interactions, thus calling for future experimental campaigns aiming at supporting improvements of MC modelling for clinical application of PET monitoring.