Tesla's radar probably just looks outward in a horizontal plane. That's what the old Eaton VORAD and most of its successors did. There are newer automotive radars which scan both horizontally and vertically [1], but Tesla doesn't seem to have those.
When I was developing Grand Challenge software, I used to have an Eaton VORAD looking out my window at home, tracking vehicles going through an intersection. It could see cars, but usually not bicycles. Range and range rate were good; azimuth info was flaky. Stationary objects didn't register because it was a Doppler radar. Output from the device was a list of targets and positions, encapsulated in Serial Line Interface Protocol.
The big problem with these radars is not seeing the road itself as an obstacle. When you're moving, everything has a Doppler radar return. Usually, the road is hit at such an oblique angle that it doesn't reflect much. But there are exceptions. The worst case is a grating-floor bridge.
LIDAR isn't a panacea. The charcoal-black upholstery used on many office chairs is so non-reflective in IR that a SICK LMS can't see it at point-blank range.
"Radar tunes out what looks like an overhead road sign to avoid false braking events"
https://twitter.com/elonmusk/status/748625979271045121