We consider strategy-proof rules operating on a rich domain of preference profiles in a set up where multiple private goods have to be assigned to a set of agents with entitlements and where preferences display satiation. We show that if the rule is in addition tops-only, same-sided and individually rational with respect to the entitlements, then the preferences in the domain have to satisfy a variant of single-peakedness (referred to as multi-dimensional single-peakedness relative to the entitlements). We also provide a converse of this main finding. It turns out that this domain coincides with the one already identified in a general set up with public goods. We relate the domain of multi-dimensional single-peaked preferences relative to the entitlements to well-known restricted domains with private goods under which non-trivial and strategy-proof rules do exist.