Maybe I should mention that I was responsible for the testing of different ATC radar systems.

I imagine the radar signature of a wingsuiter is pretty small, even if they have an exhaust-belching rocket strapped on the thing.

How well do those show up on radar?

All of the conventional radar systems installed at airports are blind to this small human jetpack due to their limited resolution.

The newer type of radar for detecting debris exist at some airports but because of their main purpose is to monitor debris on the runaway it is only for short distance and not pointed to the sky.

The main purpose of the PSR is to detect potentially dangerous flying objects which have no transponders. The PSR we tested were able to also detect e.g. parachutes and paragliders. I would assume that the radar cross section of the winged jetpack is large enough for detection by current PSR.

Depending on the angle of incidence relative to the radar, the cross-section can be quite large. But the resolution of the radar is almost certainly not sufficient to see the small object near the aircraft.

On a civilian radar, they don't. Without a transponder the tower won't see you on their screens.

https://en.wikipedia.org/wiki/Primary_radar

https://en.wikipedia.org/wiki/Secondary_surveillance_radar