Most of the time I write much the same way Matt does. I will jam to chord progressions that sound interesting to me. Eventually, subliminally, either a chorus will emerge, sometimes it is a couple of verse lines that are interesting, some times just gibberish that sounds like it might form into something. I have had complete songs come together in less than an hour and I have carried around particial songs for up to and over a year.
It seems like my biggest hang up is that I will get a complete first verse that I like and a chorus that I like and then I can't get a second, or third verse to sound as meaningful as I would like. That is when the process seem to drag on until the right words finally come forth.
I have on occasion, but not very often, written lyrics first and then set music to them. When this happens, it seems inevitable that the words will have to change a little once I figure out a melody and chord progression to go with them. It just seems like they never quite come together musically, with out the lyrics having to become singable to the way the song wants to form itself. It does n't seem like you can force them to work, they tend to kind of write themselves and go where they are meant to go.
I write differently on acoustic guitar, than I do on electric. I am not much of a piano player, but working slowly at it. I write completely differently on piano it seems. I still start with chord progressions, but they write differently for some reason. I have only written a couple of songs from a bass guitar, but bass will sometimes actually change the direction of a song as well.
The best thing I have learned over the years is not to get to stuck on any one thing, but to let it go where it wants to, even if it means completely changing direction.
I have not collaborated much on writing for the last couple of years, but am kind of getting the bug for that again as well.