We study the co-existence of star formation and active galactic nucleus (AGN) activity in Chandra X-ray-selected AGN by analyzing stacked 870 μm submillimeter emission from a deep and wide map of the Extended Chandra Deep Field South (ECDFS), obtained with the LABOCA instrument at the APEX telescope. The total X-ray sample of 895 sources with median redshift z 1 drawn from the combined (E)CDFS X-ray catalogs is detected at >11σ significance at a mean submillimeter flux of 0.49 0.04 mJy, corresponding to a typical star formation rate (SFR) around 30 M yr for a T = 35 K, β = 1.5 graybody far-infrared spectral energy distribution. The good signal-to-noise ratio permits stacking analyses for major subgroups, splitting the sample by redshift, intrinsic luminosity, and AGN obscuration properties. We observe a trend of SFR increasing with redshift. An increase of SFR with AGN luminosity is indicated at the highest L ≳ 10 erg s luminosities only. Increasing trends with X-ray obscuration as expected in some AGN evolutionary scenarios are not observed for the bulk of the X-ray AGN sample but may be present for the highest intrinsic luminosity objects with L ≳ 10 erg s. This behavior suggests a transition between two modes in the co-existence of AGN activity and star formation. For the bulk of the sample, the X-ray luminosity and obscuration of the AGN are not intimately linked to the global SFR of their hosts. The hosts are likely massive and forming stars secularly, at rates similar to the pervasive star formation seen in massive galaxies without an AGN at similar redshifts. In these systems, star formation is not linked to a specific state of the AGN and the period of moderately luminous AGN activity may not highlight a major evolutionary transition of the galaxy. The change indicated toward more intense star formation, and a more pronounced increase in SFRs between unobscured and obscured AGN reported in the literature at highest (L ≳ 10 erg s) luminosities suggests that these luminous AGNs follow an evolutionary path on which obscured AGN activity and intense star formation are linked, possibly via merging. Comparison to local hard X-ray-selected AGN supports this interpretation. SFRs in the hosts of moderate luminosity AGN at z 1 are an order of magnitude higher than at z 0, following the increase in the non-AGN massive galaxy population. At high AGN luminosities, hosts on the evolutionary link/merger path emerge from this secular level of star formation.