I personally wouldn't give permission for a story that I have wrote a significant portion of, to another person. Only if I really wanted to see it finished and I was physically/mentally incapable of doing so myself. Otherwise, I'd rather keep it stored and maybe revisited at a later date. Usually though, these stories would be completed if the Author was interested in writing period... and if they are receiving the feedback they want.. with the direction they like.
I have lost my way with a couple of stories - I don't think if I completed them people would like the direction in which they were headed. Doesn't mean I won't revisit them later or scrap the story and write the idea over again. I'm doing that right now with an idea I had back in my early e-fiction days of writing. If that goes well, I may revisit other "deleted works" here.
I would though, encourage people who want to write a similar idea to a story they like of mine, to go ahead and do so. As long as it doesn't border on plagiarism, I'd be fine with that. Mine are pretty straight forward romance ideas that are likely a dime a dozen as it is. I'd say the real trouble would be with other genres that have more strict rules and world building. I wouldn't give permission to those at all... only because a significant portion of the work is done pre-writing stage... and that I just couldn't hand over to someone else...