YES!
Would an AIDS patient want everyone knowing about their medical diagnosis? Would a cancer patient want everybody to know that they are dying?
NO!
We all want to live as close to "normal" as we can, but when word starts getting around about our medical issues, then all sense of normalcy can fly out the window. Yes, a supporter needs their own support, but that's what the internet (here) and private therapists are for. There are HIPPA privacy laws in the USA (OK, I don't know exactly what HIPPA stands for but it doesn't matter, all you need to know is that it is the acronym for the USA medical privacy laws). We deserve privacy! Once one person knows, there is NO legal recourse if they decide to tell someone else. In a small community, this can mean social, or even economic death. People may stop talking to you and you may find it near impossible to find a job. This issue goes WAY beyond the "rights" of a supporter to get help. This can make or break the life of a sufferer. Like I said, support for a supporter needs to come through the proper, PRIVATE channels.