Recently, there has been an increased interest in the study of the generation of low-energy cosmic rays (CRs; < 1 TeV) in shocks situated on the surface of a protostar or along protostellar jets. These locally accelerated CRs offer an attractive explanation for the high levels of non-thermal emission and ionisation rate, $zeta$, observed close to these sources. The high $zeta$ observed in some protostellar sources is generally attributed to shock-generated UV photons. The aim of this article is to show that when synchrotron emission and a high $zeta$ are measured in the same spatial region, a locally shock-accelerated CR flux is sufficient to explain both phenomena. We assume that relativistic particles are accelerated according to the first-order Fermi acceleration mechanism and compute $zeta$ and the non-thermal emission at cm wavelengths. We then apply our model to the star-forming region OMC-2 FIR 3/FIR 4. Using a Bayesian analysis, we constrain the parameters of the model and estimate the spectral indices of the non-thermal radio emission. We demonstrate that the local CR acceleration model makes it possible to simultaneously explain the synchrotron emission along the HOPS 370 jet within the FIR 3 region and $zeta$ observed near the FIR 4 protocluster. Our model constrains the magnetic field strength (~250-450$~mu$G), its turbulent component (~20-40$~mu$G), and the jet velocity in the shock reference frame for the three non-thermal sources of the HOPS 370 jet (~350-1000 km s$^{-1}$). Beyond the modelling of the OMC-2 FIR 3/FIR 4 system, we show how the combination of continuum observations at cm wavelengths and molecular transitions is a powerful new tool for the analysis of star-forming regions: these two types of observations can be simultaneously interpreted by invoking only the presence of locally accelerated CRs, without having to resort to shock-generated UV photons.
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