Within the city of Seattle, Seattle Animal Control and Seattle Animal Shelter are the same facility but do serve different purposes. Animal Control is there to remove dangerous animals, enforce court orders and save lost and wandering pets. Under most circumstances, these animals are held in quarantine for up to two weeks while attempts to contact the owner are made. Each time a pet arrives at S.A.C. the fines increase, and after a 3rd or 4th incident the agency can refuse to return the animal. At that point it would be turned over to the Shelter who has a large staff of volunteers dedicated to re-homing dogs, cats, etc. Euthanasia rates for the animal control/shelter are exceedingly low for a city run agency and occur primarily through court order or because of a sick/injured or highly dangerous animal.
IMHO, the best thing you could do for your neighbor is to call animal control and let the hassle and fees of getting his dog back be the motivation for his becoming a more responsible pet owner. While there are many great pit bulls out there, it's up to each owner to ensure that his is an ambassador of the breed.