Differential cross sections and photon-beam asymmetries have been measured for the gamma n --> K+ Sigma- and gamma p --> K+Sigma0 reactions separately using liquid deuterium and hydrogen targets with incident linearly polarized photon beams of E gamma = 1.5-2.4 GeV at 0.6 < cos ThetacmK< 1. The cross section ratio of sigma K+ Sigma-/sigma K+ Sigma0, expected to be 2 on the basis of the isospin 1/2 exchange, is found to be close to 1. For the K+ Sigma- reaction, large positive asymmetries are observed, indicating the dominance of K* exchange. The large difference between the asymmetries for the K+ Sigma- and K+ Sigma0 reactions cannot be explained by simple theoretical considerations based on Regge model calculations.