A thorough study of radio emission in active galactic nuclei (AGNs) is of fundamental importance to understand the physical mechanisms responsible for the emission and the interplay between accretion and ejection processes. High-frequency radio observations can target the nuclear contribution of smaller emitting regions and are less affected by absorption. We present JVLA 22 and 45GHz observations of 16 nearby (0.003=<z=<0.3) hard-X-ray-selected AGNs at the (sub)-kpc scale with tens {mu}Jy/beam sensitivity. We detected 15/16 sources, with flux densities ranging from hundreds {mu}Jy to tens Jy (specific luminosities from ~10^20^ to ~10^25^W/Hz at 22GHz). All detected sources host a compact core, with eight being core-dominated at either frequencies, the others exhibiting also extended structures. Spectral indices range from steep to flat/inverted. We interpret this evidence as either due to a core+jet system (6/15), a core accompanied by surrounding star formation (1/15), to a jet oriented close to the line of sight (3/15), to emission from a corona or the base of a jet (1/15), although there might be degeneracies between different processes. Four sources require more data to shed light on their nature. We conclude that, at these frequencies, extended, optically thin components are present together with the flat-spectrum core. The L_R_/L_X_~10^-5^ relation is roughly followed, indicating a possible contribution to radio emission from a hot corona. A weakly significant correlation between radio core (22 and 45GHz) and X-ray luminosities is discussed in the light of an accretion-ejection framework.