They commanded their own divisions, but the only dominion general to advance beyond that was Arthur Currie.
The ANZAC was commanded by Birdwood, I believe, and he was a Brit.
All other dominion units were single divisions for the most part, or doled out by the company or battalion like the ghurkas were.
Edit- Re: the Somme as a victory. It also forced the Germans to reorganize their division structure, from around 10k to around 8k--but with more heavy weapons--to keep the illusion of having superior numbers. This structure would come back to haunt them in late 1918, once they had the huge French and American divisions (around 20k each) maneuvering against them in the open. By comparison, I seem to recall that British divisions numbered around 12k throughout.
[Edited by Dhampy, 2/21/2010 9:52:09 PM]