Also known as "males."
Interestingly, I did a quick search on "who should be screeened for prostate cancer. Pretty much every result said "men" in certain age groups. this included the CDC, Mayo Clinic and American Cancer Society.
I didn't really find anything for "people with prostates." But I did find this:
https://prostatecanceruk.org/prostate-information/about-prostate-cancer
The rest of the page refers to "men."
I'm not sure "people with prostates" is a phrase used as commonly as you seem to think.
It appears to me that there is more of a push to use this inclusive language for health conditions that affect females than for health conditions that affect males. I'm not sure that's correct though, as it's pretty difficult to test.