If the second hand on the analog digital vestal Monte Carlo is not working does it need a new battery?

The Monte Carlo watch actually functions off of two separate batteries- One for the analog side (SR626SW) and one for the digital side (CR2016). It's likely that the jeweler you took it to might not have noticed that there were two batteries in there and only changed the larger, more visible digital side battery. Let the jeweler know that the analog portion runs off of a SR626SW battery which will be located underneath a black battery cushion on the analog side of the module/movement.