Yes, the traffic from a site being listed in DMOZ is indirect for the most part I'd think, unless you are a completely research/reference based site, just my guess. However, the powerful domino effect is profound and expontentially increases with the number of other directory's that pull ODP data. Anyone can get listed in Google Web, but there's only one avenue of approach to Google Directory. The "trickle down" theory continues as the baclinks increase, SERP increases, etc. I too have been contacted by webmasters who found my site via DMOZ.
I have a handful of older sites listed but there doesn't seem to be any rhyme or reason as to which ones are in and which ones aren't. At one time I worked at it a bit. Decided there were other ways the effort could be better spent.
I've submitted over 50 sites and I've got 2 in so far (that I know of). One of them only took about 3 weeks and the other took about 6 months.
It only took a few days. It didn't bring any direct boom in traffic as it only represents 1.9% of my search engine traffic. However, as others have mentioned, you will get dozens of backlinks if you are in dmoz due to other directories using it's data. Just on that alone, my site got a PR4 in the last update having only been on the radar for less than two months. I hazard a guess that the actual text that you put into the title and description during submission is quite important. Make sure there are no typos or spelling mistakes and that the description is a proper sentence.
And don't stuff either title or description with keywords. Follow the guidelines as precisely as you can. If you do keyword stuff and my time is limited I will likely skip it, as will many of my colleagues, and leave it for another day. The another day might be years in the future.
I just got one of my sites listed in DMOZ. It took 5 months. Oddly enough, that site saw no change during the Google PR update a couple weeks ago, but another site I've left untouched has moved from PR 5 to PR 6. Weird. Maybe the DMOZ backlink-goodness will kick in for me on the next Google PR update. -T