I have the exact same problem and tried the same stuff you did. Verizon iPhone 5S running 7.1. My wife has a nearly identical 5S but is running 7.0.6 and isn't having any problems. This is weird.
P.S. sundog, where are you located? As my profile suggests, I'm in the LA area.
Still working fine today in the same general area on Verizon using iPhone 5 with iOS 7.1.San Francisco Bay Area. Pretty widespread to be a "problem". Maybe just Verizon? It's weird, it's been working with 7.1 until yesterday.
even better if it's not working
Not really. For me, it's dictation I use that's not working either. iTunes matching song requests, texting, calling. I use Siri a lot so it's frustrating to me personally.
UPDATE:
Siri is now working normally along with dictation.
Not sure what was going on. Probably on Verizon's end.