I changed my trees to private simply because of this problem. I am happy to share information with genuine requests where, between us, we can agree that it is the same person/family. However, I got my own back once: when my tree was public, I entered a 'branch' that, in the end, turned out to be totally wrong. Someone who had downloaded masses of my information is still left researching an entirely 'fictitious' family since I turned my tree private and pruned the whole branch out.
The funniest thing I ever had was someone had my rellie living and dying in Melbourne, Australia, despite having downloaded all the pertinent (English) censuses etc. off my tree. When I pointed out the fact that we were talking about Derbyshire and I had, in fact, visited the cemetery where they were both buried, she wrote back and said I had to be wrong as she was correct and I should just look at her census forms in regard to this rellie!! Words fail me!