For most games, I take the "if you cross my path, you must die!" approach. Pokémon is no exception.
So yes, I battle each and every Trainer that appears in my way. The only exception are those annoying Breeders in B2&W2 who always challenge me over and over again, which I never battle more than once because their Pokémon are always the same.
"If you want to know what a man's like, take a good look at how he treats his inferiors, not his equals." -Sirius Black