In a first result, we prove that every continuous local triple derivation on a JB * -triple is a triple derivation. We also give an automatic continuity result, that is, we show that local triple derivations on a JB * -triple are continuous even if not assumed a priori to be so. These results provide positive answers to the conjectures posed by Mackey (Bull. London Math. Soc. 45 (2013) 811-824). In particular, every local triple derivation on a C * -algebra is a triple derivation. We also explore the connections between (bounded local) triple derivations and generalized (Jordan) derivations on a C * -algebra.