Drummers often get intimidated presenting tunes to a band so I thought I'd share a few thoughts to help make it go smoothly.
1) Pick your opportunities carefully:
If you're a sideman, there might not be that many chances to get your originals played.
If it's a cooperative band and you're asked to submit tunes, by all means. Don't, however, thrust your compositions on a band where they might not fit, or you would be seen as being overly pushy.
2) Have a way of communicating your tune to the others:
You either have to be able to play your compositions on piano/guitar or write out a chart for the band to read off of. I don't have enough piano chops to play both melodies and chords at the same time, so I usually write out a "lead sheet" with the necessary information on it. (I'll post an example of one of my lead sheets at the end.)
3) Learn everything about harmony you can.
Writing tunes is a chance for us to use all the information we've learned in theory classes etc. Pay attention and you'll be more likely to write something people can solo on ( in the case of Jazz compositions).
4) Go over the tunes before the rehearsal with someone you trust
In the early stages of my writing, I would often get together with the great bassist Mike Downes to get his take on my tunes. He would suggest ways of writing the chords so they would be easier to solo over, calligraphy problems etc. This helped me avoid a lot of confusion when I presented songs to the whole band. ...and with that in mind.....
5) Accept criticism with grace
If someone in the band suggests "this needs a bridge", or "this part of the progression doesn't make sense" or " this sounds A LOT like a Lady Gaga tune", they are trying to help you. You their advice (if you agree with it) to make your tunes better.
6) Make your written parts look as good as you can
A piano player I worked with years ago told me something to the effect that the better the music looks on paper (clear, isn't sloopy etc.) the more the players will treat it like important music and play their best. I think we've all experienced trying to read badly written music and how difficult it is to play it well, especially the first few times through.
7) If you want to play a lot of your own music, it's time to be a bandleader
This way, if someone really doesn't like your tunes, they don't have to be in your band!
As promised here's a lead sheet and audio example of one of my tunes.
Drummers offer a lot in compositional terms. Their rhythmic acumen, their sensitivity to feels, tempo, and mood all can lead to excellent variety in the writing department. Now let's all get writing!