(Abridged) We study relationships between the SFR and the nuclear properties of X-ray selected AGNs out to z=2.5, using far-IR data in three extragalactic deep fields as part of the PACS Evolutionary Probe (PEP) program. Guided by studies of intrinsic infra-red AGN SEDs, we show that the majority of the FIR emission in AGNs is produced by cold dust heated by star-formation. We uncover characteristic redshift-dependent trends between the mean FIR luminosity (L_fir) and accretion luminosity (L_agn) of AGNs. At low AGN luminosities, accretion and SFR are uncorrelated at all redshifts, consistent with a scenario where most low-luminosity AGNs are primarily fueled by secular processes in their host galaxies. At high AGN luminosities, a significant correlation is observed between L_fir and L_agn, but only among AGNs at low and moderate redshifts (z1). This suggests that the role of mergers in SMBH-galaxy co-evolution is less important at these epochs. At all redshifts, we find essentially no relationship between L_fir and nuclear obscuration across five orders of magnitude in obscuring column density, suggesting that various different mechanisms are likely to be responsible for obscuring X-rays in active galaxies. We explain our results within a scenario in which two different modes of SMBH fueling operate among low- and high-luminosity AGNs. We postulate, guided by emerging knowledge about the properties of high redshift galaxies, that the dominant mode of accretion among high-luminosity AGNs evolves with redshift. ; Comment: 18 pages, 9 figures, 3 tables