I have been out of the classroom for a few years but even my small district had at least something to fill out in regards to privacy. Recently I attended a substitute orientation for a moderately sized district. We had a form to complete with quite a few types of personal information listed that we had to check if we did not want the information released. That information included home address, mailing address, all telephone numbers, veteran status, date of birth, former name, race and ethnicity, marital status, information on family members, tax information, spouse's name, and social security number. We were told that anything we did not check had to be released to anyone who asked due to open records requirements (USA). This is why I recommended asking about a form to complete so that personal information can be closed. If she wants to leave some information open that is fine but my list above gives an idea of what a district in the US is required to give out if requested.
I do understand that the directory was something the district created rather than this being a situation of just anyone asking for information. Either way, there is probably some way to opt certain information out for the future.
She is not being dramatic for not wanting certain information to be published. I recommend though that she calmly checks on how to prevent release in the future since it is too late for this year's directory distribution.