You somewhat misinterpreted what @jerv actually said, @2davidc8. It’s not that there is “some leakage” to the public of radioactive material or waste at nuclear power plants, but there is always “some amount of radiation”.
It is literally and physically impossible for us, given current technology, to shield ourselves from all electromagnetic radiation. That is why we are all exposed to cosmic radiation, for example – radiation from stars that mankind will never approach, including the Sun – and we’re exposed to all of the radiation coming from the core of our own planet as well. Whether you know if or not and no matter where you are, you’re exposed to radiation from the Rockies in the Western USA, and to radiation from uranium deposits in Africa, too, as well as Chernobyl, Fukujima, Three Mile Island and every safely operating power plant in the USA, France and the rest of the world, not forgetting all of the nuclear submarines operating in the world, wherever they may be. You’re exposed to all of that, and you can’t avoid it. If there is life around Alpha Centauri then that is exposed to – and receiving – some of our radiation, too.
You can increase the shielding between yourself and those sources, and you can increase the distance between yourself and those sources and you can decrease the time you spend in close proximity to high radiation sources such as Xray machines, transcontinental flights, trips to the mountains and… coal mines and buildings made of concrete.
For those who have a pathological dread of “radiation is bad; it’s killing us”, that may seem like bad, bad news. But most scientists and doctors who have studied radiation and its health effects – good people who are not employed as shills for the nuclear power industry, the defense industry, the mining companies or the government – have determined that the health effects of the low-level background radiation that we’re all exposed to are nil. (Some suggest that it’s actually beneficial.) Almost no one with any credibility – no one that I know of, in fact – says that “all radiation is dangerous and bad” and we must increase our shielding.
Furthermore, because of the stochastic nature of radioactive decay, if you took a radiation detector into your home today – where it would “read” background radiation even if you live in the safest house in town – there’s no way of knowing whether the random indications that reader would give are emissions from Alpha Centauri, the Sun, your grandpa’s radium-dial watch, or Chernobyl or a submarine on the other side of the planet. The reader would give more-or-less random indications that “there’s a wave” and “there’s a wave” and “there’s a wave” – but there would be no way for you to tell (in the case of background radiation) where those waves originate.
Now, if the radiation detector starts to indicate “more than background radiation”, then an investigator would track that to see whether there might be something in or around your house that’s emitting more than should be expected. Radon gas coming from beneath your foundation is often suspect, and in some parts of the country it is expected – and dealt with. Even some of the house contents can be more radioactive than normal, including the concrete of the foundation. If it’s the power plant next door, then the plant owner will have an especially great interest in locating and containing the source of any radiation that you may be experiencing.
I think that in @jerv‘s generally excellent write-up, he might have more accurately stated that “radiation workers” at power plants “may receive” 5–6 times the annual radiation dose of the average American, but that’s just because they so frequently work so much closer to extremely high levels of radiation inside the plant itself. And that 5–6 times is only measurable on devices that they’re required to wear to monitor their exposure; you can’t tell and they can’t tell by any health consequences, “Oh, look at the effect all that radiation has had on him” at the end of his life. You can’t tell, and doctors can’t tell.
The annual levels of exposure that the NRC permits to radiation workers are calculated as a fractional amount of an expected lifetime dose where “we still can’t predict any certain – or even likely – harm from this amount”. People in the plant who don’t work with radiation directly, including office workers and other non-technical people, don’t have any more exposure than you or I – and right now I live about 60 miles from the nearest operating plant.
The biggest danger to living near an operating nuclear power plant – a much larger danger by far, in fact – is the likely media-driven panic that may occur among ignorant people if an event of some kind were to occur. That’s a danger worth thinking about, in fact. So if you’re one of those people who would panic over something that you don’t know anything about, then it would be a good idea to stay away from that plant. (But don’t stay in a cave, because the natural radiation occurring from materials inside and beneath the cave might be worse than being in the parking lot of the nuke plant.)
People who understand radiation and its effects do not fear it unreasonably, though they do respect it. Be one of those people.