Is it safe use half-eaten bones from a family dinner meal to make stock? Or is this unsanitary, and it's better to just obtain bones through filleting and deboning while raw or after slow or pressure cooking?
It's not sanitary, in the sense of following the health rules. Especially since it's unlikely that you're following the two-hour guidelines: the gnawed bones have been in the danger zone enough to potentially pick up an enterobacter that produced heat-stable toxins. Boiling will not fix that. And having been in somebody's mouth increases the chance that such bacteria is one that infects humans.
Consider it this way: even if you're not squicked out by the basic concept, how would you feel if the bones had been left out on the plate for a day? A week?
As for whether it's safe, the short answer is "no".
The odds it being actually dangerous are pretty low. The food safety rules are designed to keep the most at-risk people safe: small children, people with compromised immune systems, etc. Given how many other opportunities there are for food-borne infections to be picked up, I'd consider something that I picked up and simmered for many hours to be about as free from pathogens as anything I got out of the dirt, i.e. vegetables, which we often eat raw.
So I'll admit to having done it, but I wouldn't feed it to anybody except myself.
@Joshua Engel covers the safety and sanitary aspect very well.
But if you are keen to give it a go, perhaps you can roast the bone remains first...
Boiling won't kill all the germs, neither will roasting - but it will kill more than boiling.
If you have good heat in the oven (180 - 200C) for a good 30 minutes or so (longer if you turn the heat down so they don't burn), they will probably develop a better taste when boiled for stock afterwards.
Throw in some veges for the last 20 minutes, and also add them to your stock