A few years ago, a position became available within our organization. At the time, I worked very closely with 3 other people, and I did consider them friends (albeit, work friends) -- they were at my wedding, etc. My coworkers and I all met the job requirements, and it was a significant promotion. After a little bit of encouragement from my supervisor and DH, I applied, and was pretty open about the fact that I was doing so. A few days later, one of my co-workers came up to me with a kind of sheepish look on his face and told me that he, too, had thrown his hat in the ring. It was clear that he felt awkward about it ((I think his words were, "I think I might have done something that you won't like."), and I assured him that I didn't expect everybody else to step back just because I wanted the position. We went through the process together, we took the "exam" sitting next to each other (silently), and wished each other luck when we crossed paths in the parking lot as I was leaving my interview and he was entering. I didn'thelp him study for the exam or prep for the interview because we were competing for the same position, and of course I'd prefer to get the job over him, but I certainly didn't harbor any ill will toward him for wanting the same thing I did.
On the other hand, one of my other co-workers said absolutely nothing about the job opening. She didn't come to work on the day of the exam, and left the office at lunchtime without a word, the day after my interview. I found out later that she had applied for the job as well (though I had suspected).
I felt very different about my two co-workers after that situation -- while of course the second co-worker had every right to apply for that job, and she probably had her reasons for not wanting to disclose that information to me, it still just didn't sit quite right with me.
So, long story short -- while I don't think it's rude for your friend to ask you about the details of your job application, OP, I *do* believe it's a little rude if he's not being honest (even if just by omission) about why he wants those details. I like the suggestion from a PP who said that you should say, "Why do you ask?" which just kind of forces his hand.