No need to restrict it, was debating with myself whether or not a distinction needed to be made.
Originals, I've seen it go various ways - with a songwriting duo where the drummer and I were sidemen ("if you want to write songs, go form another band" - I appreciated the honesty), to the more collaborative approach I'm used to these days with The Inevitable Teaspoons - where someone will bring in anything from the beginning of an idea to a mostly finished song and we all get to work on refining our parts and make suggestions about others. The collaborative approach is slower, but it's enjoyable, and it's a hobby so we're not exactly bothered if we only release an album once ever 6 years.
In the covers band, we're all free to make suggestions, and while we all have a veto we don't tend to use it unless the song is obvious cheese, we try to think about songs from an audience point of view. I do play a couple of songs I can't abide, but it's not about me, the audiences love it and entertaining them is the primary reason for us doing it in the first place. Most suggestions get at least tried - sometimes they fall by the wayside because they sound naff in a single guitar situation, sometimes they'll get as far as a gig and fall flat on their faces, other times they'll naturally drift out of the setlist because we notice they're not doing so well or because we've got bored playing it, or replaced it with something better that does a similar job. There's no tinpot dictator or central figure - we all work together for the good of the show. We're as busy as we want to be so I'd like to think that the strategy works.