Peregrines are very similar to Sparrowhawks and the photo is not very clear on the website, hence my concern about the leg covering. Peregrines are larger than Sparrowhawks and have similar markings on the front and both have yellow eye rings. The Peregrine has covered legs but the bird in the photo has an uncovered right leg. Both have yellow legs and feet. The Sparrowhawk has a whitish top to its beak and a pale line over its eye, which ties in with the photo. The Peregrine has a yellow top to its beak. Peregrine catch their prey in flight and generally take their kill back to the nest.
In answer to your further question, it is possible you have a nest nearby. Generally the male Sparrowhawk will do the hunting and, being smaller than the female, will not take something as large as a pigeon. It will often have a favourite post/area where it will take the prey, which will be quite close to a nest. It will then pluck the kill and often call the female down to collect. I would expect that any young have fledged by now so perhaps she is just feeding herself.