I don't think it has anything to do with socialisation, more guarding her property.
You could start training her to ignore the other dog. Each day go out and either wait for other dog to come outside or organise with that owner to let dog out at certain time and try to refocus Pixie from the other dog, ie when dog comes out use treats to get Pixies attention, make her sit then reward her, make her 'look' at you and reward her, do this a number of times then go in side. If you can do this everyday, it may help to desensitize her interest in the other dog. Also important for you to completely ignore the other dog, she will follow your lead.
This is what I did with the dog that lives behind us, my 2 ignore him when he is in his back yard and they are in ours with only a fence separating them, funny though when he walks past our front yard (our property is fully fenced) on lead with his owner both Lola and Harry carry on like pork chops.