When you speak of 'persons own insurance', what do you mean? If you organise an outing and I go along, what insurance would you expect I would have to cover an accident? Do you mean cover if I caused an accident to another person, or do you mean cover if I was involved in an accident myself, such as I simply fell and broke my arm?
It is possible, though by no means certain, that if I cause an accident to another person, my home insurance might cover my legal liability. If I fall and break my arm, I have no insurance of my own to fall back on.
These are very simplistic examples.
If the organiser of an outing could face a liability for an injury, then the organiser should consider insurance. Now the organiser, might be simply a person collecting money to bring a group of people on a bus tour. In that case, (again very simple) if the bus had an accident, the motor insurers would be involved. Once the people leave the bus and enter a hotel for a meal, the hotel liability policy comes into play.