  We point out how future lepton colliders can provide unique insight into the
scalar sector of TeV scale models for neutrino masses with local $B-L$
symmetry. Our specific focus is on the TeV scale left-right model, which
naturally embeds this $B-L$ symmetry. In particular, we make a detailed study
of the lepton collider implications of the neutral ($H_3$) and doubly-charged
($H^{\pm\pm}$) scalars from the right-handed triplet Higgs that is responsible
for the spontaneous breaking of the $B-L$ symmetry and implementing the seesaw
mechanism. Due to mixing with other scalars, the neutral scalar $H_3$ could
acquire sizable flavor violating couplings to the charged leptons. Produced
on-shell or off-shell at the planned $e^+e^-$ colliders, it would induce
distinct lepton flavor violating signals like $e^+e^- \to \mu^\pm \tau^\mp ~
(+H_3)$, with the couplings probed up to $\sim 10^{-4}$ 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 $H^{\pm\pm}$ 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 $H^{\pm\pm}$ like
$e^+e^- \to H^{++} H^{--},\, H^{\pm\pm} e^\mp \mu^\mp,\, \mu^\pm \tau^\mp$, the
Yukawa couplings can be probed up to $10^{-3}$ 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.
