Actually, it's not legal privacy concern. Since HIPAA is the main player in this field, we'll use that as the example. HIPAA permits ANY release of PHI when used in the course of patient care. Telling a crew where to go and who to find is part of patient care. Obviously you can try to avoid releasing the name, but there is no legal requirement.
Just like you can talk to an RN in a room and not be afraid of being fined because a patient in the next room over-hears, you don't have to be afraid to say PHI over the radio and getting in any sort of legal trouble.
Think about it... the law is loosely defined as providing any info to someone else that can identify a patient. Dispatch can withold the name, but saying "A 49 year old female at 123 Main Street for vaginal bleeding", it can be easily determined who the patient is, and therefor is a breach of HIPAA... if it weren't already considered legal.
And that's if a dispatch center is even considered a covered entity in the first place, which many are not.