If someone eats meat that was not properly slaughtered or food that was cooked on a grill on which bacon is also cooked, as is the case in most American restaurants, and that person then eats with a group of people one or two hours later, does the impurity transfer from their mouth to the food given that they put the food in their mouth and we all eat from the same plate? Is their mouth impure, and does the food therefore become impure by them eating for the same plate as us? Or does their saliva clean their mouth.

All perfect praise be to Allah, The Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallahu ‘alayhi wa sallam, is His slave and Messenger. 

It is clear that your question is the result of some kind of Waswasah (obsessive satanic whisperings). In brief, you cannot judge a specific person's mouth as being impure unless you know with certainty that he has eaten the flesh of a dead animal or that an impurity came in direct contact with his mouth. When a person eats the flesh of a dead animal, for instance, his mouth becomes impure and he is obliged to wash it. However, the impurity of his mouth does not necessitate the transfer of the impurity to the food that he is sharing with others.

If we were to assume that what you mentioned is true and that a person with an impure mouth ate with a group of people, then the food that they are eating does not become impure unless it is known with certainty that it has come in contact with the impurity. It is known that the food that enters the person's mouth and comes in direct contact with his impure saliva is the food that only he eats, and not the other food on the plate. So disregard such Waswasah and pay absolutely no attention to it, for dwelling on it leads to much evil.

Allah knows best.