We present J-band imaging and H+K-band low-resolution spectroscopy of 2MASS 1207-3932AB, obtained with VLT NACO. For the putative planetary mass secondary, we find J = 20.0 ± 0.2 mag. The HK spectra of both components imply low gravity and a dusty atmosphere for the secondary. Comparisons to synthetic spectra yield T_eff,A ≈ 2550 ± 150 K and T_eff,B ≈ 1600 ± 100 K, consistent with their late-M and mid- to late-L types. For these T_eff and an age of 5-10 Myr, evolutionary models imply M_A ≈ 24 ± 6 M_Jup and M_B ≈ 8 ± 2 M_Jup. Independent comparisons of these models to the observed colors, spanning ~I to L', also yield the same masses and temperatures. Our primary mass agrees with other recent analyses; however, our secondary mass, while still in the planetary regime, is 2-3 times larger than claimed previously. This discrepancy can be traced to the luminosities: while the absolute photometry and M_bol of the primary agree with theoretical predictions, the secondary is ~2.5 ± 0.5 mag fainter than expected in all bands from I to L' and in M_bol. This accounts for the much lower secondary mass (and temperature) derived earlier. We argue that this effect is highly unlikely to result from a variety of model-related problems and is instead real. This conclusion is bolstered by the absence of any luminosity problems in either the primary or AB Pic B, which we also analyze. We therefore suggest gray extinction in 2M1207B, due to occlusion by an edge-on circumsecondary disk. This is consistent with the observed properties of edge-on disks around T Tauri stars and with the known presence of a high-inclination evolved disk around the primary. Finally, the system's implied mass ratio of ~0.3 suggests a binary-like formation scenario.