As someone who does R&D testing on plastics that are used in medical devices, I have some insight. Of course the type of plastic matters, but all plastics use carcinogenic chemicals during the manufacturing/extrusion process.

To make most plastic, a polymer resin is mixed with additives such as solvents, plasticizers, and stabilizers at high temperatures. Ideally, you want the additives to evaporate out during production so that you're left with just the newly formed plastic.

But some of these additives get trapped in tiny air pockets between polymer chains. When they're reheated, the polymer chains relax and release the volatile, carcinogenic additives into the air.

This is likely where the toxicity is coming from, not the polymer chain itself. So regardless of the type of plastic used, reheating the polymer during 3D printing will release some volatile additives.