I have to say, though, I'm leaning a little more towards only trainers. Although, people could have their pokemon talking to each other normally, with the trainers still present, and then put translations of what they're saying underneath or something.
That could work as well.
Just absolutely no talking like they're in the show. They make animalistic noises, not say their name over and over like they got hit in the head with a cinder block one too many times.