No ticket was issued? Does our insurance really HAVE to pay?

The other poster, Entidtil, is right. The police are only concerned with road law. If they are unsure of whether a violation of the law has occurred, they can't charge you for it. On the other hand, your insurance company needs to work out who was most at fault, or who created the most dangerous situation, that led to the accident. As this was a "controlled" intersection - that is, it had a stop sign, the expectation would be that on entering the intersection, the driver faced with the stop sign must avoid causing a collision with vehicles already on that road. Whether you saw the other car or not is not really an issue - that's what most accidents are caused by - some element of misjudgement. So, if another vehicle collided with yours, it would indicate that the way was not clear. As it was your responsibility to be sure the way was clear before entering the intersection, and as the other vehicle hit you in the side while you were physically still inside the bounds of the intersection, you would be held most at fault. So, yes, your insurance will pay for it.