Why doesn't Jonas tell his friends about his training in The Giver?
Jonas doesn't talk about his training with his friends, because he has been ordered not to.
After all of the children find out what role they will be serving in the community, the children all receive folders that contain instructions about their new role. Jonas's folder contains a list of nine instructions. Instruction number four is quite clear that Jonas is not allowed to discuss with anyone what his training involves.
Do not discuss your training with any other member of the community, including parents and Elders.
The other instructions tell Jonas other things that he finds more immediately disturbing. For example, Jonas is shocked that he is exempt from the rudeness rule and that he is now allowed to lie. As Jonas begins and continues in his training with the Giver, he learns to understand rule number four. His friends and parents wouldn't understand anything that he is learning anyway. They do not have Jonas's gift.
The Chief Elder has told Jonas that although his position is a great honor, his training will be alone and apart. And, when Jonas receives the instructions for his assignment, they contain 9 important directions. Number 4 clearly states, "Do not discuss your training with any other member of the community, including parents and Elders" (Chapter 9). Jonas also realizes that, even if he were allowed to tell, his friends, who do not have his gift, would never understand what he was talking about. What he is learning is too far out of their own experience.
Yes, it is a part of the rules and regulations of his training not to talk about his memories with anyone. Also, Jonas quickly realizes that the people of his society are not equipped to deal with the pain that Jonas is receiving as a side-effect of his training.