Most often, I come along with the bass lines/chord sequences etc for a song which everybody works out a part for (or sometimes I suggest), and singer writes lyrics.
A couple of times our singer has come along with lyrics which we'll then work out riffage for.
Either way, we jam it until it works.
With a cover we'll look for a spin on the idea like funkin' or punkin' itup, or making it into a mash-up. I object to doing straight covers!