While the double end bag does help with punching accuracy, I guess, I'm not sure how much. I used to be able to hit it for a round while carrying on a conversation and never looking at it. Using a peanut bag, real small, was challenging at first...and I was far from a prodigy.
To develope accuracy, it is something you work at all the time, in every aspect of your training. You put marks on your heavy bag, and that is where you land your punches; when you are practicing slipping a jab and, for example, landing a straight right to the heart and a left hook to the liver, you hit those marks, not just 'the bag.' When your trainer is working your on the mitts, he should be giving you realistic targets to hit.