Police departments vary wildly as to who they hire to do what. When I started, "communications" personnel were all commissioned officers. Often, they were temporarily "off the street" due to injuries or whatever, but there were many who just enjoyed the job and were good at it.
Now, it's fairly common to have civilians working "the desk", with perhaps a police officer or sergeant supervising to field the constant flow of legal questions.
( I have intended to write a book about my experiences working "the desk"....I have stories..)
Dispatchers, the guys who actually work the radio, are almost always civilian. It's a dreadful job, frankly. Highly stressful and with a load of responsibility. Those that are good at it are worth their weight in gold. Those who are not are a danger to everyone and far too often get retained anyway as it's so hard to find people to fill these typically-underpaid positons.