An ovarian cyst is not an infection, so don't worry that it will somehow spread. Losing one ovary does not affect the other, except that the other will now ovulate every month (instead of every other month) to fill in for the missing ovary.
It would be unusual for her to have a problem with the remaining ovary. Usually, ovarian cysts pose no other issue once they have been removed. She should talk to her doctor to make sure there was not anything unusual about her cyst that could affect the other ovary. But, in general, she should be fine!