I agree that it should be reported to the agency she works for. I don't really have a problem with them talking about it in front of her, as it sounds like she's a home care nurse. My aunt has MS, and she has a nurse come to help her a few hours a few days a week to give her mother a break. The nurse is generally "around" when they're conversing, but that doesn't mean that they can't talk about things that they're going to do together. It's a little different from the rules you'd have at, say, work or school, or when socializing with somebody.
I can also see them not getting that the nurse was really saying she wanted to come, although in retrospect that's obviously the point when they should have clarified. But I could easily see the phrasing making them think that she was simply saying that it was nice of her to do this dinner, and that if she were one of the relatives, she'd have liked to come (in other words, that she thought it was a nice idea), or just a joke that fell flat.
As for how the nurse knew when and where, she could probably have seen it on a calendar or information for the elderly woman (after all, she had access to the house and the information would probably be somewhere easily accessible for the woman's sake), or she could have asked the older woman about it and the older woman answered, thinking the nurse was just curious, not that she was planning to come.