I'm not going to share job-hunting information with a friend
Oh is that true? Why would you not want to share job-hunting information with your friend?
I think it's better if I do the job-hunting on my own
You are afraid that he will get a job before you and I think it's greedy and selfish behavior. Seem like this might taint the friendship if he finds out this is what's on your mind. It really doesn't cost you anything to help and share this information, so you should try to be a good friend to him.
I'm not going to share anything.
Don't you want to avoid possible future fighting though? I don't think it would be such a problem to let them know how your job hunt is going.