We report studies of charge injection and transport in ambipolar, predominantly n-type, and unipolar p-type single, strongly quantum confined PbSe nanowire (NW) field effect transistors (FETs). The PbSe NW FETs operate as Schottky barrier FETs in which the Fermi level is pinned near midgap, consistent with the low ionicity of PbSe, and is nearly invariant with semiconductor doping. Electron and hole mobilities increase monotonically with decreasing temperature, dominated at high temperature by electron-phonon scattering with no evidence of scattering at low temperatures. Transport in NWs is consistent with their single crystalline nature. Surface oxygen used to dope the NWs acts remotely, providing a promising route to dope nanostructures.