Agriculture is a presumed permission, so no planning permission needed to keep livestock.
when buying - there may be restrictions on what you can do, generally if buying there will be no problem, but get your solicitor to check covenants just in case
renting will generally have restrictions on what you can and cannot do.
There are no rules on distance between so you could keep them on the fence line, and in theory you can put them your side of your neighbours fence !
However you will always come under nuisance, which will generally fall into 3 categories - 1. smell -outdoor low intensity pigs don't smell, but neighbours like it as an excuse. 2. Noise - young pigs will normally only make a noise at feed time, so feeding promptly after you arrive at their pen will mean this is at worst a couple of minutes. 3. vermin - pigs eat their food at a sitting, so there should be no leftovers to attract rats, but make sure feed is kept in vermin proof containers and don't ad-lib feed. Flies can be attracted to dung, but generally that means they are eating and getting rid of it, so taking away pig poo smell, and if the flies are with your pigs, they are not causing a nuisance to your neighbours.
Assuming you are being considerate, then if neighbours complain to the council, they will send someone, and you just need to show that your pigs are not causing any issues.
I would try and site the pigs away from sight and hearing of neighbours if possible, and that way you can immediately say that you have considered them, and in any case your are far less likely to have an issue. Many people who have been on our courses have set up in large gardens, and neighbours have never know they keep them until pork is offered !