As far as I understand it, an ozonator is only effective when the water is flowing. So a circulation pump is necessary for optimal performance of the Ozonator.
I have a 40-watt equivalent circ pump in my Hotspring Jetsetter. My insulation keeps the tub so quiet I can't even tell it's on if it were not for the large glowing logo. Note - my tub is also located less than 3 feet from my bedroom wall so sound levels were a major concern to me. Fortunately the circ pump is very quiet.
Note the disadvantage is that this means your ozonator is running 24/7 which depending on it's hour rating, works out to around 2 years of use before needing replacement. The Circ pump is also another item to fail in the tub eventually. Personally I feel the benefits of an ozonator outweigh the negatives, but that's my opinion.