A double-walled carbon nanotube (DWNT) provides the simplest system to study the interaction between concentric layers in carbon nanotubes. The inner and outer walls of a DWNT can be metallic (M) or semiconducting (S), and each of the four possible configurations (M@M, M@S, S@S, S@M) has different electronic properties. Here we report, for the first time, detailed Raman spectroscopy experiments carried out on individual DWNTs, where both concentric tubes are measured under resonance conditions, in order to understand the dependence of their electronic and optical properties according to their configuration. Interestingly, for the three DWNTs that were studied, the inner-outer tube distance (e.g., 0.31-0.33 nm) was less than the interlayer spacing in graphite. We believe these results have important implications in the fabrication of electronic devices using different types of S and M tubular interconnects.