The black hole-and-galaxy (BH-galaxy) co-evolution paradigm predicts a phase where most of the star formation (SF) and BH accretion takes place in gas-rich environments, namely, in what are likely to be very obscured conditions. In the first phase of this growth, some of the galactic gas is funnelled toward the centre of the galaxy and is accreted into the supermassive BH, triggering active galactic nucleus (AGN) activity. The large quantity of gas and dust hides the emission and the AGN appears as an obscured (type 2) AGN. The degree of obscuration in type 2 AGNs may even reach values as high as N H > 10 24 cm −2 (i.e., Compton-thick, CT). Population synthesis models of the X-ray background (XRB) suggest that a large population of CT-AGN is, in fact, needed to explain the still unresolved XRB emission at energy above 20 keV. In this work, we investigated the properties of 94 Ne  V 3426 Å-selected type 2 AGN in COSMOS at z = 0.6 − 1.2, performing optical-to-far-infrared (FIR) spectral energy distribution (SED) fitting of COSMOS2020 photometric data to estimate the AGN bolometric luminosity and stellar mass, star formation rate, age of the oldest stars, and molecular gas mass for their host-galaxy. In addition, we performed an X-ray spectral analysis of the 36 X-ray-detected sources to obtain reliable values of the AGN obscuration and intrinsic luminosity, as well as to constrain the AGN properties of the X-ray-undetected sources. We found that more than two-thirds of our sample is composed of very obscured sources ( N H > 10 23 cm −2 ), with about 20% of the sources being candidate CT-AGN and half being AGNs in a strong phase of accretion ( λ Edd > 0.1). We built a mass- and redshift-matched control sample and its comparison with the Ne  V sample indicates that the latter has a higher fraction of sources within the main sequence of star-forming galaxies and shows little evidence for AGNs quenching the SF. As the two samples have similar amounts of cold gas available to fuel the SF, this difference points towards a higher efficiency in forming stars in the Ne  V -selected sample. The comparison with the prediction from the in situ co-evolution model suggests that Ne  V is an effective tool for selecting galaxies in the obscured growth phase of the BH-galaxy co-evolution paradigm. We find that the “quenching phase” is still to come for most of the sample and only few galaxies show evidence of quenched SF activity.