We describe the use of partially overlapping galaxies to provide direct measurements of the effective absorption in galaxy disks, independent of assumptions about internal disk structure. The nonoverlapping parts of the galaxies and symmetry considerations are used to reconstruct, via differential photometry, how much background galaxy light is lost in passing through the foreground disks. Extensive catalog searches and follow-up imaging yield ~15-25 nearby galaxy pairs suitable for varying degrees of our analysis; 11 of the best such examples are presented here. From these pairs, we find that interarm extinction is modest, declining from AB~1 mag at 0.3RB25 to essentially zero by RB25; the interarm dust has a scale length consistent with that of the disk starlight. In contrast, dust in spiral arms and resonance rings may be optically thick (AB>2) at virtually any radius. Some disks have flatter extinction curves than the Galaxy, with AB/AI~1.6 this is probably the signature of clumpy dust distributions. Even though typical spirals are not optically thick throughout their disks, where they are optically thick is correlated with where they are most luminous: in spiral arms and inner disks. This correlation between absorption and emission regions may account for their apparent surface brightness being only mildly dependent on inclination, erroneously indicating that spirals are generally optically thick. Taken as an ensemble, the opacities of spiral galaxies may be just great enough to significantly affect QSO counts, though not enough to cause their high-redshift cutoff. Based in part on archival observations with the NASA/ESA Hubble Space Telescope (HST) obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555.