Would a 911 dispatcher work out of a police station?

It happens every which way, probably. In Louisville, Mississippi (a small town) the 911 building is just behind the police station. I don't know if they have officers manning the phones or paid county employees.

In my locale (suburban-rural county seat), dispatchers aren't police officers, but they do work out of the police station. I interviewed for a position (didn't get it), but it didn't include becoming a police officer.

Yes.
The small town I worked for in AZ had the dispatch room in the police station.

Different jurisdictions handle it differently. In some, the EMA (Emergency Management Agency) runs 9-1-1, which is responsible for scrambling all emergency responses. In others, it could the Sheriff's Office or the county police. In the smaller rural communities around here (I live in South Georgia), dispatchers aren't necessarily POST-certified.

As a former reporter/editor for a daily newspaper, I'd call and visit law enforcement regularly, and I still routinely work with law enforcement in my current position as a Public Health public information officer (Public Health is also a first responder).

Hope this helps. Good luck with your writing.

From what I understand, 911 is a paid central service. Towns and cities contract to different services. It is in their benefit to have higher collaboration, so they pay for cost. If you call 911 you want it directed to the right town/city, no matter where you call from.

A dispatcher usually is in a police station as a police employee (of course, rural areas might have to combine) - not necessarily a trained or uniformed officer. They send out officers, field calls and visitors to the police station and handle incoming 911 calls (from the service). You can always call your station directly (usually there are at least two numbers- emergency and non-emergency) or call 911 and be wired to the service (who would contact your local dispatcher if necessary).