As someone who volunteered for the AZD12222 trial and who has various "co-morbidities", I consulted with 2 of my doctors beforehand. Given that none of the vaccines being tested are live virus vaccines (meaning there is 100% no way to get COVID-19 from the vaccine, unlike say, the polio vaccine), and all had already passed Phase 1 safety trials, they both put it as (paraphrasing): "It's a personal choice, but if you do it, thank you for helping science."
I think COVID-19 is perhaps different than many other types of diseases in it's mainly-selective dangerousness to select groups of people. A healthy 30-something is unlikely (but possibly) to have severe, hospital-landing COVID. Someone with various co-morbidities, however is more than twice as likely to die than someone without. As such, I think that makes it much more likely that this group will volunteer, than "normal" people. It's certainly what drove me to do so.
The only issue they seemed to really care about during the intake was immunosuppression. As another comment pointed out, as long as other medical conditions were stable, you were eligible. From the vaccine makers point of view, I'd surmise this group is almost desired (my personal opinion), as you're less likely to a have asymptomatic COVID infection, which given how they are running the trial are unlikely to get noticed.
I take great solace in hearing that among the people who got COVID even when having received the vaccine (I'm talking about the AstraZeneca one, which is ~70% effective), there were no serious cases that required hospitalization. This indicates that even among the "not-effective" group, the vaccine confers some immunity. For me, that bring the personal risk factor from "possible death sentence" to "just like the flu...it'll suck for 2 weeks but I'll be fine".
Obviously I don't know if I got the vaccine or not (though given I felt crappy the next day, I'm fairly sure I did), which as another commenter points out is what makes these trials effective. Neither I, nor the doctors I met with (nor anyone at the clinic where I received it) has any idea if I got vaccine or placebo. In fact, they even had the vaccine couriered over from another hospital pharmacy using only a numeric identifier so really nobody knows what I got.
Coming back to the original question, I think many people, especially with co-morbities might seek out advice before enrolling, but at least in my case, two doctors were all for it.
I'm thankful I enrolled, and I'd encourage others, especially those who are more likely to get severe COVID to do so as well!