In this sequel to a companion paper, we complement our analysis of the binary projections of the International Trade Network (ITN) by considering its weighted representations. We show that, unlike the binary case, all possible weighted representations of the ITN (directed/undirected, aggregated/ disaggregated) cannot be traced back to local structural properties, which are therefore of limited informativeness. Our results highlight that any topological property representing only partial information (e.g., degree sequences) cannot in general be obtained from the corresponding weighted property (e.g., strength sequences). Therefore the expectation that weighted structural properties oer a more complete description than purely topological ones is misleading. Our analysis of the ITN detects indirect eects that are not captured by traditional macroeconomic analyses focused only on weighted rst-order country-specic properties, and highlights the limitations of models and theories that overemphasize the need to reproduce and explain such properties.