Although now routinely incorporated into hydrodynamic simulations of galaxy evolution, the true importance of the feedback effect of the outflows driven by active galactic nuclei (AGN) remains uncertain from an observational perspective. This is due to a lack of accurate information on the densities, radial scales and level of dust extinction of the outflow regions. Here we use the unique capabilities of VLT/Xshooter to investigate the warm outflows in a representative sample of 9 local (0.06 $<$ z $<$ 0.15) ULIRGs with AGN nuclei and, for the first time, accurately quantify the key outflow properties. We find that the outflows are compact (0.05 $<$ R$_{[OIII]}$ $<$ 1.2 kpc), significantly reddened (median E(B-V)$\sim$0.5 magnitudes), and have relatively high electron densities (3.4 $<$ log$_{10}$ n$_e$ (cm$^{-3}$) $<$ 4.8). It is notable that the latter densities -- obtained using trans-auroral [SII] and [OII] emission-line ratios -- exceed those typically assumed for the warm, emission-line outflows in active galaxies, but are similar to those estimated for broad and narrow absorption line outflow systems detected in some type 1 AGN. Even if we make the most optimistic assumptions about the true (deprojected) outflow velocities, we find relatively modest mass outflow rates ($0.07 < \dot{M} < 11$ M$_{sol}$ yr$^{-1}$) and kinetic powers measured as a fraction of the AGN bolometric luminosities ($4\times10^{-4} < \dot{E}/L_{BOL} <1$%). Therefore, although warm, AGN-driven outflows have the potential to strongly affect the star formation histories in the inner bulge regions ($r \sim$ 1kpc) of nearby ULIRGs, we lack evidence that they have a significant impact on the evolution of these rapidly evolving systems on larger scales.