Conduction in thin random networks of single-walled carbon nanotubes (SWNTs) is typically dominated by metallic SWNT segments and limited by variable-range hopping (VRH) in disordered junction regions. However, in our surfactant-free networks, we show that in parallel with VRH there is another mode of conduction involving both semiconducting and metallic SWNTs. This second process showing activated behavior makes a substantial contribution to conductance at higher temperatures, with similar activation energies in different samples despite a large variation in overall magnitude of the conductivity. From the magnitude of the activation energy (0.160.05\,eV), we ascribe this additional term to activation across Schottky barriers between metallic and semiconducting SWNTs in conducting paths involving a small number of semiconducting SWNTs. The Mott parameters for VRH are also independent of the overall conductivity, indicating that the local structure of intertube contacts between metallic SWNTs in the most conductive paths is also similar in samples of different overall conductivity.