The Cortana apps for Android and iPhone will allow users to do many, but not all, of the same spoken- and/or typed tasks as they will be able to do on Windows-based devices. Microsoft is working to enabled the syncing of Cortana's notebook across all devices where the personal assistant is available.
On the Xbox, users seemingly will need a Kinect sensor in order to use voice commands to talk to Cortana. I had wondered whether users also might be able to speak into a headset to direct Cortana, hands-free.
"The best way to use Cortana on Xbox One is with Kinect. Voice commands make hands-free navigation easy so you can stay focused on your game. If you want, you can type questions and fill in search bars using your controller, as well," said a Microsoft spokesperson in response to my query. (I've asked again if "best" means "only," but have yet to hear back.)
Update: I'm still not sure if Kinect is required, given this rather oddly-worded response to my follow-up question from the same spokesperson:
"We want to enable the intelligence and power of Cortana for all customers across a wide set of devices. What we announced is that Cortana with Kinect on Xbox One delivers great gaming specific capabilities. We have no further details to share at this time."