As a writer, I think my biggest fear (aside from fear of the dark, fear of black widow spiders, fear of clowns...etc) is having new people say, "Hey, can I read what you've written?"
My first instinct is to think "YES!!! A NEW PROOF READER!", and I get really excited... Until I print something off, or go to click the "Send" button through e-mail. It's all fun and games until that point. Right then, I feel crippling anxiety take over, "Oh, God, what if they don't like it?" "What if they find out I'm not as good as I think I am?" Then there's always the fear of, "What if they try to steal my work...my ideas?" (Which has actually happened. I wrote my friend a poem, and he stole a line from it and put it into his story WITHOUT asking. And then proceeded to say "I thought we wrote that together". Yea. No.)
When I send these materials, I find myself constantly apologizing and saying things like, "Let me know if there's anything you don't like." This is (in reality) a good thing to say, I want to edit, I want my work to be more readable to others. But at the same time, I end the e-mail with, "You may not like this...but".
Now--put this in perspective for literary agents or literary magazines. The query and cover letters are supposed to say "This is what my novel/piece is about..." and "I am strong and confident with my work." You're not supposed to apologize. You're not supposed to allow yourself the weakness you feel. You're supposed to expect that they'll love your writing and want to print you...as soon as possible.
Maybe I get more anxious with people I know reading my work (especially Creative Nonfiction...as there are some stories people who know me, just shouldn't know). Maybe I'm afraid of what they'll say about it...Thankfully, most everyone who has read what I've written is constructive with their criticism and willing to read the re-write. They're not afraid to say "This is weak, this is lacking, I am not engaged with your narrator". Some also say, "This line is awesome. I totally know what you mean."
The writing world is terrifying to me. But just as I mentioned being constantly afraid of speaking in public in this blog, I fight through my fear. Every. Day. I. Can. It can only make me a better writer.