There's no one person that has this job. There's a small portion of the population able to see Mushi, and a percentage of those people decide to make a living out of dealing with them. Most Mushishi, at least once in their career, have to mediate between a more intelligent mushi and the affected human population.

If things are more separated as it seems to be for your world, it should be noted that the mushi don't exist in a separate world. They get a lot more personally involved.