For churches and fire stations listed on the ACA maps, my experience is excellent. They almost always let me sleep inside.
For other churches that I have just approached when I got there, my experience is mixed. In many cases, there won't be anybody there; I call the number on the sign out front; if nobody answers, I leave a message; the call usually isn't returned, or isn't returned soon enough to do me any good. If you happen to find somebody at the church (e.g., the pastor, the cleaning lady, etc.), your chances of success are much better. More often than not, they let me stay. Anybody but the pastor usually has to check with the pastor first. I did spend an hour at one church, they called the pastor and the pastor came to the church just to tell me no; that was odd. In many other cases, the pastor has said yes. I usually end up sleeping on a couch in the bible-study room or day-care area, or on the gym floor.
I have just camped without permission when I couldn't find anybody to talk to. It has almost always worked out fine. I've slept on the porch of a NFS office, and they invited me in for coffee when they arrived in the morning. I've pitched my tent behind a police station and they didn't bother me, even after they noticed me there in the morning. I slept in the park across the street from a police station, and they brought me food and firewood when they saw me over there. I've slept in many city parks and left a note on the bathroom door asking them not to lock it for that night, which they have always honored.
There's always a few uncomfortable moments when you do something like this, because you don't know how they're going to react.