I have been teaching writing for over 15 years and it never ceases to amaze me at how many talented writers I have come across over the years. On the first day of a course you can usually pick out the good writers, the great writers and the extraordinary writers.
The good writers will raise their hands, offer to read out loud and on occasion answer a question posed. A great writer will offer to read out loud, share their experiences and share some of the disappointments they have had as a writer. However, it is the extraordinary writers who I will have to solicit to read, and who then hesitantly prefaces it by saying, "It really isn't very good," then read it so fast that it is hard to keep up with what they are saying. After the rest of the class comments about how good the writing is, I turn to the writer and ask them. "You were embarrassed to read that?" They will giggle and say, "yeah I didn't think it was as good as what everyone else read."
This is not uncommon, since one of the typical characteristics of writers is that we all tend to be perfectionists. If not forced to hand in an assignment, they will often edit and edit and edit some more. If we left it only up to writers, there would probably be no books to read. They would all still be in the hands of the writer being edited and edited and edited.
Trust me, all good works of writing need to be edited and edited again, and then read by at least one other set of eyes before considering sending it off for publication, but there comes a time when like with a child you must understand the need to pack it up and send it on its way.
So how do you know if you are a good writer? Read other writers. Read your work outloud and don't be afraid to send your work out for submission. Don't be discouraged if you get a few rejections, all writers do. Have confidence and write every single day. Even if you start out as a great writer you too can become extraordinary!