From your short description, it could be "Voyage from Yesteryear" by James P. Hogan.
World War III is about to break out, so a planned automated probe to be sent to Alpha Centauri is modified to carry a genetic database of humans in an attempt to make sure that humanity survives even if WWIII kills everyone in Earth.
WWIII does occur. The USA and other countries survive, and decide to find out what happened to the first probe.
Most of the story is told from the point of view of the people and crew of the ship from the USA.
By the time the Earth has recovered enough to send the new missions, the people on Alpha Centauri have developed a new society, guided by the robots from the first ship.
It is a liberal, post scarcity society. All basic needs (food and housing, etc.) are provided to all people by automated factories. The people spend all their time doing whatever interests them, and getting better at it.
There's no money. "Rich" people are rich in recognition - if you are good at something, people know and respect you for the abilities you have developed and the knowledge you have gained.
There's no government as such. People organize things as needed, and cooperate to do the best for themselves and society.
The new settlers from Earth discover all of this one piece at a time.
The new settlers intend to take over by force, but the people already on Alpha Centauri have had decades to plan how to convert the newcomers to their way of life without fighting any full out battles.
The plan works. The Alpha Centauri folks subvert and convert nearly all of the newcomers peacefully. Only the hard core of the worst the Earth has to offer - its leading politicians and army officers - resists and tries to start a war.
By this time, there are only a few of that sort left. They attempt an attack with nuclear weapons. The Chironians (the planet is named Chiron) destroy the missiles with a directed antimatter weapon, and defeat the last hardliners from Earth.
The story ends with the Chironians awaiting the coming of the next ship from Earth (sent by China) and the expectation that they will subvert its crew to a better way of life as well.
Your added details make it sound even more like this is your book.
There are no official police on Chiron, but there are descriptions of incidents in which ordinary (and not so ordinary) people act to defend the peace and other people - up to and including shooting an aggressor dead if needed.
The "stores" are as you describe them - walk in and pick up what you need (or the robotic "salesman" talks you into taking.) The system works because the people are responsible and only take what they need for things they are doing or working on.
All that matters is that whatever you do, you do it to the best of your ability. By the standards of a Chironian, a good house painter is as valuable as a surgeon - it is a recognition of the ability and work that goes into being good at what you do. Do what you like, do it well, and you will be respected.
A very good book, that I enjoyed very much.