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In order to use the camera to sign you in, Kinect “measures distances between key points on your face to create a numeric value that represents only you.” This information stays on your console and is not shared. The value itself is stored as a “very long set of numbers” and, according to Microsoft, “[n]o one could look at the numbers and know they represent you.”

For gameplay purposes, Kinect will map the distances between your body’s joints to create a stick figure. The numeric values for your stick figure are temporarily stored in memory on your console during a gameplay session but are destroyed when you quit the session.

If you’re playing online “Microsoft may collect those numeric values to enable and improve gameplay and improve the gaming experience” but the values sent to Microsoft are “destroyed after analysis is complete.” The company notes you will not be able to be identified by your stick figure representation.
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In terms of voice chat, however, the privacy statement advises users they “should not expect any level of privacy concerning your use of the live communication features such as voice chat, video and communications in live-hosted gameplay sessions offered through the Services.” Microsoft says it may monitor communications “to the extent permitted by law, but we cannot monitor the entire Service and make no attempt to do so.” The company reminds users that other players could record and use your communications on Xbox Live and your “[c]ommunications in live-hosted gameplay sessions may also be broadcast to others.”
Welp. Fuck this shit, man.