According to the Star Trek Star Charts book, the Dominion's territory in the Gamma Quadrant is about 14,000 LY in diameter. Nearly twice as large as the Federation.
We saw in episodes like "The Quickening" what happened to world's that defied the Dominion. Rest of the time it just seems like a "Your world is a Province of the Dominion now. We'll come by every once in a while for tribute or if we need something your world can offer us. Just don't rebel and it's okay."