AEH is spot on. My thoughts are if your son is getting what he needs at this moment at his current school, just go with it. If you don't think so, its best to figure out why. Can the school do more with the resources available? Do they feel they should do more for your child? Can you supplement after school? Transitions are hard for kids and the emotional/social part of "schooling" is just as important (imo)so I recommend taking that into consideration. Also, some gifted schools focus too much on making parents happy ("my child goes to a gifted school") rather than focusing on what the child needs educationally. Is your child high achieving or does he do simply what is required? Is he motivated by grades or a drive to learn more? If you do consider the gifted school as your friend suggested, find out more about the type of instruction they provide. How much homework is required? Is there differentiation within grade or do they support acceleration if necessary? Does your son do better in some subjects than others?
These are all questions I wished I had known to consider when we found out that our twins were gifted. The folks on this board have been amazingly helpful with guidance along this crazy journey. Best wishes