  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]
