I have been so frustrated by this Siri issue for the past two months that I went ahead and downloaded the iOS 9.3 beta too. I am so pleased to say that It seems to have resolved the issue, as far as I can tell. I can immediately call up music playlists, have my new messages read back and dictate replies, and many of the other functions that were reportedly not working. If you are looking for an immediate fix, and don't want to wait for the new official upgrade, this could help you too.
Now, let me say this: Could this issue be related to the apps on my phone or my settings, possibly interfering with the Siri service or connection to the server? I have wondered because my wife's iPhone 6S 64gb, activated at the same time with the same iOS version as 128gb model, has never had this problem. So I began to wonder if there are other security features or apps that could be impacting this. Just something I hope will help others determine why this is occurring infrequently.
I also found that iOS 9.2.1 (and it's beta update) did not resolve anything. My guess is that when the release of 9.3 comes along (estimated as this Spring, possibly as early as March), this issue will be fixed. Here are the resources that I used to learn about what we can expect in 9.3 and how to upgrade to the 9.3 beta now:
iOS 9.3 Preview: http://www.apple.com/ios/preview/
iOS 9.3 beta Installation Instructions: https://beta.apple.com/sp/betaprogram/guide
An unofficial account of Installing iOS 9.3 beta in more detail: http://www.imore.com/how-download-ios-93-public-beta