As has been said, it depends on the type of bird as to whether you can shoot it.
It is strongly advised not to shoot anything with a pistol, as it will likely not kill it unless you were very very close to the bird.
Lastly, make sure the pellet does not leave your boundaries if you are going to do it, if you can keep within these criteria you should just buy an actual Air Rifle, get some practice in and then take them out cleanly.
I have had similar problems with woodpigeons, myself and the neighbours like to take them out when the opportunity presents itself.