I work in medical research, currently with kidney stones, and patients do get a say now. Even a stone surgically removed (basically medical waste like you said) can't be used without permission. We have to get consent to study the stone and tell them what we plan to do with it. We also ask a separate permission for tissue banking and future experiments that we may think of. But we aren't going to make billions of dollars off patients like in the HeLa situation. A patient can revoke their consent at any time. We would just stop using their sample from that time forward. Studies, consents, and protocols go through committee reviews and audits. If someone or an institution is found to be misusing patients or their samples, labs can be shut down or even all clinical research at a certain institution.
Tl;dr: We have to get a multi-page consent signed to be able to just see what a kidney stone is composed of (consent for actual medical waste)
I don't think a similar situation like HeLa could happen now (making billions from using something without permission) Someone and/or some place would get sued for some of that money I'm guessing and places shut down.