We study the co-existence of star formation and AGN activity in X-ray selected AGN by analyzing stacked 870um submm emission from a deep and wide map of the Extended Chandra Deep Field South, obtained with LABOCA at the APEX telescope. The total X-ray sample of 895 sources with median redshift z~1 is detected at a mean submm flux of 0.49+-0.04mJy, corresponding to a typical star formation rate around 30Msun/yr for a T=35K, beta=1.5 greybody far-infrared SED. The good S/N permits stacking analyses for subgroups. We observe a trend of star formation rate increasing with redshift. An increase of star formation rate with AGN luminosity is indicated at the highest L_2-10>~1E44erg/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. This suggests a transition between two modes in the coexistence 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 star formation rate 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. The change indicated towards more intense star formation, and a more pronounced increase in star formation rates between unobscured and obscured AGN at highest luminosities suggests that luminous AGN 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. [Abridged]
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