Coarsening droplet patterns in a two-dimensional binary mixture of amphiphiles, confined to a monomolecular film at an air-water interface, have been investigated via fluorescence microscopy and extensive digital pattern analysis. We focus here on structural aspects of the disordered patterns formed as a result of off-critical, isothermal surface pressure quenches. Structural correlations manifest themselves in the shape of the scaled droplet radius distribution, P(R/(R)). "Packing" constraints, amplified by a substantial excluded volume effect, predominate in the densest patterns and, at an area fraction of minority phase or "coverage" phi similar or equal to 0.25, produce a virtually Gaussian shape of P(R/(R)). Patterns of lower coverage, with phi less than or similar to 0.15, tend to produce skewed shapes, as expected on the basis of droplet-droplet interactions. The initially uniform spatial distribution of droplets is observed to be unstable to the eventual formation of large-scale inhomogeneities. The possible connection of this phenomenon to the presence of electrostatic interactions is discussed.