Using deep Herschel and ALMA observations, we investigate the star formation rate (SFR) distributions of X-ray selected AGN host galaxies at 0.5 < z < 1.5 and 1.5 < z < 4, comparing them to that of normal, star-forming (i.e., “main-sequence”, or MS) galaxies. We find 34–55 per cent of AGNs in our sample have SFRs at least a factor of two below that of the average MS galaxy, compared to ≈ 15 per cent of all MS galaxies, suggesting significantly different SFR distributions. Indeed, when both are modelled as log-normal distributions, the mass and redshift-normalised SFR distributions of X-ray AGNs are roughly twice as broad, and peak ≈ 0.4 dex lower, than that of MS galaxies. However, like MS galaxies, the normalised SFR distribution of AGNs in our sample appears not to evolve with redshift. Despite X-ray AGNs and MS galaxies having different SFR distributions, the linear-mean SFR of AGNs derived from our distributions is remarkably consistent with that of MS galaxies, and thus with previous results derived from stacked Herschel data. This apparent contradiction is due to the linear-mean SFR being biased by bright outliers, and thus does not necessarily represent a true characterisation of the typical SFR of X-ray AGNs.