I think they are definitely one good way of encouraging songwriting. I believe it helps when the competition gives a theme or some indication of the style of song the judges will be looking for. It's also helpful where it is possible to hear the other entries or at least the ones that have been successful. An annual competition, of course, gives a clear goal and plenty of time to write a song and fully review it before submission.
However, there are also plenty of songwriting courses and songwriting circles of different types. I know of 3 songwriting circles in my own area alone (Birmingham UK). One of the features I have noticed is encouraging participants to produce material quickly by for instance agreeing a theme with the challenge to produce a new song before the next meeting. I can see the point of this in terms of building discipline but it is not an approach that works for me personally.
I guess another way is to find a songwriting buddy. It worked for The Beatles.