I heard it said (somewhere, I don't remember where, and I think the person who said it was quoting someone else) that every writer starts with fanfiction. I don't think that's literally true (though I must admit my first stories--dictated to my mom and then carefully illustrated with crayon--were in essence My Little Pony fanfics), but it wouldn't surprise me if most writers start with stories that are at least less than skillfully done homage.
For ex, one of the first long stories I wrote was a complete rip-off of Robin McKinley's THE BLUE SWORD. I didn't take *everything*, but I took all the story elements I loved--discovering a hidden talent within oneself, the training stuff, the coming to understand and even love a people you thought were the enemy--and back then I didn't know how to take all that stuff without taking large portions of plot as well. So if you were to read that story, you could easily pinpoint the moments that are exactly the same as the moments in THE BLUE SWORD, just with different characters and a different setting and (obviously) different wording.
By the time I wrote my first novel, I'd gotten a fair bit better. But I still did some significant borrowing. The three main characters, though different from their namesakes in all but name, were Sorsha (I have loved Willow a long time), Rydia (via Final Fantasy IV), and Pathlan (via my mangling of a name in one of the DEATH GATE CYCLE books). I also stole a device from a DEATH GATE CYCLE book in order to make a fade-to-black love scene happen. And in general, the thing was a mass of genre cliches.
With both stories, I could see right after finishing them that I'd been too lazy, relied to much on my reading and not enough on my own imagination, and each was banished to the far corners of my hard drive. But they were still good practice, in writing in general, in figuring out what I wanted to write about, in building a plot and characters, and all sorts of good stuff like that.
That was thirteen years ago, and I've come a pretty long way (I hope!). I think the longer you write, the more you get to know how to make the story your own, even if you're inspired in certain ways by other stories, or you're drawing in techniques you learned while reading other people's books. You learn how to make something a homage, or to twist it until it's completely your own, with just the feeling you wanted to keep. But I suspect most of us start writing because we love reading, and so what we're reading can't help having a pretty heavy influence on those first efforts.
Anyone else have embarrassing book-borrowing tales to share? :D