"The Dominion does not own the entire Gamma Quadrant. We had explored the GQ for two years before encountering the Dominion, so it's not as though the wormhole opens up in their living room. There are other races in the GQ that are not part of the Dominion and the Ferengi at least have established trade with some of them. When the Dominion told us to stay out of the GQ, it was as if China told the US to stay out of the Yellow Sea. China is the big boy in this neck of the woods, and you better take their warning seriously, but at the same time we have trading partners and allies there and hey, freedom of the seas and all that."
That's straight from Moore's mouth.
My guess is that the Dominion pretty much ignores anyone that isn't important to them. They rule the GQ as much as the Federation rules the AQ, that is not at all.
That's my take on it too. A quadrant is still a really big chunk of real estate when you take into account all three dimensions, and even an empire that encompasses an area of thousands of light-years only occupies a percentage of it. If you encounter the Dominion, yeah, you're going to have to deal with them but there are likely many races within the Gamma Quadrant that may not even have heard of the Dominion just like how there are races in the Alpha Quadrant that haven't heard of the Federation, IMO.