Question about immunity and virus transmission

[Double-Helix, BSN, RN]

I was thinking about this recently, since I work with some immunocompromised children and preventing new infections can be a challenge. Many of our patients have brothers and sisters at home that come to visit.

Here's my question: If you have already had a disease (let's say the chicken pox) are you become exposed to it again, can you still pass it on to others? So if a sibling (who has been vaccinated) is exposed to chicken pox by a student at school, and comes to visit his sister (who has not been vaccinated) can he spread the chicken pox to the patient?

ETA: I'm thinking that you cannot pass on the virus if you are immune, because otherwise it would make no sense for us, as healthcare workers, to have titers drawn to prove we are immune. If we took care of a patient with a contagious disease (say an ER nurse cares for a patient with chicken pox before she realizes what the child has), we would have to avoid being around other patients for a certain period of time.

I feel like I should remember this from school... but I'm not positive. I tried looking it up online but I couldn't find any reliable websites that answer this specific question.