There is no "right" way to treat or educate a gifted child. If you like the school, your son is happy, and the school is happy that he is achieving well and not getting bored, subdued or disruptive, then that's just fine. If the situation changes and he starts to get bored, or starts being criticised for, eg, not paying attention, then do something. This doesn't sound to be the case for you at the moment, so it's probably enough to ask to be kept informed and to be given any advice on what you could be doing at home to help support his work at school.
In terms of what the school may already be doing, in my ds1's (state) school, the children identified as gifted in maths or reading do get an extra hour a week in each subject, doing more challenging work which I believe they can then build on within the normal classroom environment if they have the ability to work independently (ie get on with it...). If this is not sufficient or doesn't seem to be working, then they may give some one-to-one time wherever possible. My ds1 gets a couple of hours a week of one-to-one tuition (most weeks) and comes home ever so happy, talking about the fun things he's done. We have not pushed for any of this, the school has provided it of its own choice because they appear to know my ds very well and have been very responsive and sensitive to his needs, which is fortunate, because he is the sort of child who would seriously underachieve if not given extra support. He also loves his time in the normal classroom, albeit he is rather quiet (which is what pushed them to provide extra help, as they could see he wasn't doing what he was capable of) - everything goes in, and there is so much more to learn at primary school than reading, writing and arithmetic!
If your ds does start to have problems in class, then his giftedness may be considered by the school to be a special need for which extra classroom support is required. The state sector is not devoid of the ability to deal with gifted children, although what extra support you get will depend on the level of motivation of the school and individual teachers, parental support and the caseload of any outside experts that may be called in to give advice on how to challenge an unusual child that is seen to be exceptionally bright but for some reason under-achieving in the school environment.
Certain members of both my and my husband's family have been hothoused in the past - eg moved into the private sector and put into a class of children several years older than them. In one case, this appeared to work (except that said child was expected to play sport with older children, too, despite being utterly unathletic), but in the other cases it resulted in one life blighted by low self esteem and alcoholism and in the other, by a long-term anger at his mother for sending him to a school miles from home, surrounded by older children who teased or bullied him and which deprived him of friendships with children near his own home and of his own age. He was successful academically and in his career, but still has a massive downer on his own mother!!! She forgot that a child needs help developing his social skills and self-confidence just as much as his academic abilities. Plenty of gifted children don't actually achieve much in later life as a result of this failure to recognise that pushing them hard in one area of their development may have an unintended and undesirable effect on other areas of their development.
So, I would say, don't assume your ds is bored or underachieving, or being let down in some way by not being sufficiently challenged. Act on the basis of his actual behaviour, and if he seems well adjusted and generally happy, then don't rock the boat too much for now!!