To get the ball rolling...
For me:
I always start a poem on paper. It is a very deliberate act for me. I usually start with one line or a specific metaphor in mind. That is the first grain of sand. In the case of my most recent the grain of sand was the common phrase "it'll never fly". I put it up against the idea of an individual's flag. Whether it is your freak flag, geek flag, political opinion flag, your cause, your badge of courage. Those things we're all made of symbolized in fabric as a banner.
Soon as the idea seemed to have some merit after letting it and associated ideas roll around together in my head for a day or two and I realized I could turn that cliche phrase into something different - I got out my notebook. I created the character, and the pieces of the flag from there.
So the "on paper thing". Paper always happens first for me with poetry. And it's odd, because with fiction I always use a keyboard. But poetry, at least for me, requires a slower, and more intimate, relationship with the words. Each one seems to matter more and what the idea generates seems to flow more organically with pen in hand.
So I write what I know, which rarely has usable structure. I scribble all over it, rearrange parts, write in margins, then use that version as an outline and start on a fresh page, this time pulling the key elements to the forefront. Key pieces stand out at this moment and show themselves to be the bones of the piece.
Then I got through it again, this time strengthening the bits that work, killing the bits that don't. That's building the muscle for me. It's the flesh around the bones.
Next, I type it. It always changes from paper to screen, in the typing process I edit as i enter it.
The fine-toothed comb edits happen on screen. I read it several times and make minute changes to detail.
Then I post, at which point I absorb feedback, read it on my own a dozen more times and tweak structures, weak points, eliminate useless words.
Sometimes, as with something I wrote a while back called Clutches, I never got past the feedback and into what felt final. I did multiple paper versions, tweaked it mercilessly in pixel form, and still never felt like I had actually boiled it down to its best potency. (that one still bugs me). In the end, I was not able to use a scalpel for the fine edits, and did a really lousy hack job on the thing. It lost its melody to my ear and I couldn't really get a grip on it again.
Other times, something transforms from lots of chicken scratched, red-lined words into an actual poem.
some of the finer points of my process:
I often keep, for each individual piece, a little list of ideas I want to use to support the over-arching theme, metaphor. Some of them are phrases - others are just words which I believe are able to be symbols in the piece. The one that get used are the ones which can carry a metaphor, or provide a symbol, or even support a nice heavy double life for the piece.
Brunette Laundress was like that with it's last line "she folds". Those two words carried an enormous amount of poetic weight, and they needed to have the flexibility to be interpreted in more than one way.
Sometimes the word choice of one line would change the entire poem for me. Sort of like basing an entire outfit on the merit of an adorable pair of shoes.
Of course, lots of other things go into it. The musicality of the words, once the skeleton is on paper has everything to do with what gets chopped, rearranged or rewritten, and which parts of it thrive.
Ultimately, no matter how long it actually takes (one hour or three weeks) by the time it actually shows up for comment on AW, I've typically scribbled up multiple pages of notebook paper, and read the various versions dozens and dozens of times. I've edited it multiple times in a word doc, before I ever share. Once it is here, I might change it a dozen more times, because somehow, the imperfections show in the blue light of AW more sharply than they show at my own house. (something about that reminds me of the skinny mirrors in department stores, but we'll let that pass).