We study a simple left-right symmetric extension of the Zee model for neutrino mass generation. An extra SU(2)L/R singlet charged scalar helps in generating a loop-induced Majorana mass for neutrinos in this model. The right-handed neutrinos in this case are very light of the order of a few electron volts to a few mega-electron-volts, which makes this scenario quite different from other left-right symmetric models. We have analyzed the scalar potential and Higgs spectrum in detail, which also play an important role for the neutrino phenomenology. We identified the parameter regions in the model which satisfy the experimentally observed neutrino masses and mixings along with other experimental constraints. We then studied the collider signatures of the charged scalar at e+e- colliders with different benchmark points. It is possible to get a huge enhancement in the production cross section of the charged scalar at a lepton collider compared to the hadron colliders, resulting in a much stronger signal, which can be easily observed at the upcoming International Linear Collider or Compact Linear Collider experiments. © 2018 authors. Published by the American Physical Society.