We use the LLAMA survey to study the density and outflow rate of ionized gas in a complete volume limited sample of local (<40 Mpc) luminous (43.0 < log LAGN(erg/s) < 44.5) AGN selected by very hard 14-195 keV X-rays. The detailed data available for this survey enable us to measure the density of the outflowing ionized gas in the central 300 pc of these AGN using three different and independent methods (the standard [SII] doublet ratio; a method comparing [OII] and [SII] ratios that include auroral and transauroral lines; and a recently proposed method based on the ionization parameter). For each method there is, as expected, a modest spread of densities among the AGN in the sample. But remarkably, the median densities for each method differ hugely, by an order of magnitude from below 400 cm-3 to almost 5000 cm-3. We discuss how the derived densities can be reconciled, and what the impact is on the implied outflow rate.