I should have come back to this, but I forgot.
This is the new behavior of Siri, at least on newer iPhones. The reasoning is that the beep was there to let you know when to start speaking. Siri is now fast enough on many devices that you can now start speaking as soon as you press the home button. On those devices, you won't get a beep. If you have vibrate turned on in Settings > Sounds, you still get a little buzz; otherwise Siri is silent, and ready immediately (iPhone only, iPad's don't have a vibrate motor).
I don't have an iPad Air 2, so I can't confirm on that device. The iPad Air still beeps. My iPhone 6 Plus does not. On the other hand, my iPhone 6 Plus still has enough lag that need to wait just a bit to talk. Not much, but at least until the Siri screen appears.
Anyway, it's not a bug. It's intentional. If you still need a bit of feedback, try enabling vibrate. I do. It's all the feedback I need, once I realized that the beep was gone.