We point out how future lepton colliders can provide unique insight into the scalar sector of TeV scale models for neutrino masses with local symmetry. Our specific focus is on the TeV scale left-right model, which naturally embeds this symmetry. In particular, we make a detailed study of the lepton collider implications of the neutral () and doubly-charged () scalars from the right-handed triplet Higgs that is responsible for the spontaneous breaking of the symmetry and implementing the seesaw mechanism. Due to mixing with other scalars, the neutral scalar could acquire sizable flavor violating couplings to the charged leptons. Produced on-shell or off-shell at the planned colliders, it would induce distinct lepton flavor violating signals like , with the couplings probed up to for a wide range of neutral scalar mass, which is well beyond the reach of current searches for charged lepton flavor violation. The Yukawa couplings of the doubly-charged scalar to the charged leptons might also be flavor violating, which is correlated to the heavy right-handed neutrino masses and mixings. With a combination of the pair, single and off-shell production of like , , , the Yukawa couplings can be probed up to at future lepton colliders, which is allowed by current lepton flavor data in a large region of parameter space. For both the neutral and doubly-charged cases, the scalar masses could be probed up to the few-TeV range in the off-shell channel.