It can be transmitted in the air or from direct contact or from someone's hands or clothing. It's best to keep rabbits in a different room, and as stated previously, ALWAYS wash hands before handling the chins after handling the bunnies.
There are other diseases that rabbits can pass on, but Pasturella is the one big one that chin breeders worry about. It can be deadly to a chinchilla and extremely difficult to cure, if it even can be cured. As Alicyn stated, good hygiene is a great place to start. A lot of people have rabbits and chinchillas and they do fine. When the chins do well they are usually in a completely different area than rabbits.
Recently I had a woman call me wanting with her chin with Pasturella "in its eye". The chin had be placed in the same area in her home as several rabbits and they were allowed to play. The chin came from a rabbit breeder that allowed for her chins and rabbits to share the same bunny barn. All the chins there died of Pasturella and the "breeder" still sold several chinchillas despite knowing they were infected, and all of her rabbits eventually succumbed to Pasturella and passed away. That is an extreme case there, but it illustrates the point that chins and rabbits should not be kept in close quarters and also precautions must be made when keeping the two species in the same household.
