Nuclear starbursts and active galactic nucleus (AGN) activity are the main heating processes in luminous infrared galaxies (LIRGs) and their relationship is fundamental to understand galaxy evolution. In this paper, we study the star formation and AGN activity of a sample of 11 local LIRGs imaged with subarcsecond angular resolution at radio (8.4 GHz) and near-infrared (2.2 mu m) wavelengths. This allows us to characterize the central kpc of these galaxies with a spatial resolution of similar or equal to 100 pc. In general, we find a good spatial correlation between the radio and the near-IR emission, although radio emission tends to be more concentrated in the nuclear regions. Additionally, we use an Markov Chain Monte Carlo code to model their multiwavelength spectral energy distribution (SED) using template libraries of starburst, AGN and spheroidal/cirrus models, determining the luminosity contribution of each component, and finding that all sources in our sample are starburst-dominated, except for NGC 6926 with an AGN contribution of similar or equal to 64 per cent. Our sources show high star formation rates (40-167 M(circle dot)yr(-1)), supernova rates (0.4-2.0 SN yr(-1)) and similar starburst ages (13-29 Myr), except for the young starburst (9 Myr) in NGC 6926. A comparison of our derived star-forming parameters with estimates obtained from different IR and radio tracers shows an overall consistency among the different star formation tracers. AGN tracers based on mid-IR, high-ionization line ratios also show an overall agreement with our SED model fit estimates for the AGN. Finally, we use our wide-band Very Large Array observations to determine pixel-by-pixel radio spectral indices for all galaxies in our sample, finding a typical median value (alpha similar or equal to -0.8) for synchrotron-powered LIRGs.