Abstract We present sensitive 870 μm continuum measurements from our ALMA programmes of 114 X-ray selected active galactic nuclei (AGN) in the Chandra Deep Field-South and Cosmic Evolution Survey fields. We use these observations in combination with data from Spitzer and Herschel to construct a sample of 86 X-ray selected AGN, 63 with ALMA constraints at z = 1.5–3.2 with stellar mass >2 × 1010 M⊙. We constructed broad-band spectral energy distributions in the infrared band (8–1000 μm) and constrain star-formation rates (SFRs) uncontaminated by the AGN. Using a hierarchical Bayesian method that takes into account the information from upper limits, we fit SFR and specific SFR (sSFR) distributions. We explore these distributions as a function of both X-ray luminosity and stellar mass. We compare our measurements to two versions of the Evolution and Assembly of GaLaxies and their Environments (EAGLE) hydrodynamical simulations: the reference model with AGN feedback and the model without AGN. We find good agreement between the observations and that predicted by the EAGLE reference model for the modes and widths of the sSFR distributions as a function of both X-ray luminosity and stellar mass; however, we found that the EAGLE model without AGN feedback predicts a significantly narrower width when compared to the data. Overall, from the combination of the observations with the model predictions, we conclude that (1) even with AGN feedback, we expect no strong relationship between the sSFR distribution parameters and instantaneous AGN luminosity and (2) a signature of AGN feedback is a broad distribution of sSFRs for all galaxies (not just those hosting an AGN) with stellar masses above ≈1010 M⊙.