  Galaxy mergers are likely to play a role in triggering active galactic nuclei
(AGN), but the conditions under which this process occurs are poorly
understood. In Paper I, we constructed a sample of spatially offset X-ray AGN
that represent galaxy mergers hosting a single AGN. In this paper, we use our
offset AGN sample to constrain the parameters that affect AGN observability in
galaxy mergers. We also construct dual AGN samples with similar selection
properties for comparison. We find that the offset AGN fraction shows no
evidence for a dependence on AGN luminosity, while the dual AGN fractions show
stronger evidence for a positive dependence, suggesting that the merger events
forming dual AGN are more efficient at instigating accretion onto supermassive
black holes than those forming offset AGN. We also find that the offset and
dual AGN fractions both have a negative dependence on nuclear separation and
are similar in value at small physical scales. This dependence may become
stronger when restricted to high AGN luminosities, though a larger sample is
needed for confirmation. These results indicate that the probability of AGN
triggering increases at later merger stages. This study is the first to
systematically probe down to nuclear separations of <1 kpc (~0.8 kpc) and is
consistent with predictions from simulations that AGN observability peaks in
this regime. We also find that the offset AGN are not preferentially obscured
compared to the parent AGN sample, suggesting that our selection may be
targeting galaxy mergers with relatively dust-free nuclear regions.
