Dear Miss Snark,
How does a writer know if he / she is any good?
You don't. And it doesn't matter. "Good" is a term we throw around a lot but it's meaningless mostly.
I always say write well, but that's just to get out of the slush pile and to get you to quit obsessing about margins and times new roman. What I look for is work I can sell. Mostly that's "good" but not always.
If you really have no sense of whether your work is lucid, clear, compelling or cogent and persuasive take a writing class. First, it's fun, and second, most colleges have writing labs where you can take your term papers or essays and some poverty stricken graduate student will snark you into improving. I learned a LOT about good writing in undergraduate school from those guys.
Lots of the commenters will suggest a crit group:
A critique group is only as good as the people in it, and you'd be surprised how much crap comes out of crit groups. I got a huffy letter from querier once (back in the days of trying to be helpful---long gone I assure you) telling me my advice to seek out a crit group was clearly stupid cause she'd been in one for years. I got a good laugh from that. If you're in a crit group and people tell you need a crit group after they see your writing, you need a new group.
If you need to hone your sense of good writing, cause the best way to know what's good is to read good stuff, read the ALA's good books. Forget those idiotic best seller lists, go for the librarian list everytime. And of course, read all the books that Miss Snark tells you too.
And then of course, when you're brave, Elektra stands ready at the Crapometer Annex.