How are people working in psychophysics trained?
Short answer
Psychophysics is a multidisciplinary field featuring people with a Physics, Psychology, Medicine and/or Biology background. 

Background
OK, here follows a lot of personal account to illustrate where psychophysicists may come from.

Traditionally, psychophysics is a part of Psychology. 

I do a lot of psychophysics and I'm a Biologist by training (MSc), but learned psychophysics during a postdoc. My supervisor teaching me psychophysics was a physicist (MSc) who acquired their skills during their PhD (in visual sciences).   

My current supervisor is a physicist by training (MSc), but has done heaps of psychophysics as part of his post-grad Audiology training.

Many of my psychophysics colleagues are in fact physicists.