For example, let's say you try this with 5 pairs of students after school for one hour. Each day you measure how well it went on a 1 to 10 scale. If the first few weeks it's a 4 out of 10, you try little solutions to get it to 6, 7, 8 out of 10. Trial and error. Rewards and consequences. Using personal adult-to-kid relationships so that the kids try harder because they want to impress/please the adult.
Do that until you prove you can make tutoring work outside of class.
Then you probably need to find which teacher finds this most intriguing, and try to bring it into the normal class. You say to the teacher "Let's experiment with this in your class. Instead of your normal lecture, do a very short lecture and then divide students into pairs."
You are transparent with the kids. You say something like:
"We are interested in having you children work together in pairs sometimes during class. We think you will learn more and enjoy it more. We are going to give you a survey. We'd like to get your opinions on the matter. Who would like to try to learn this way? Write down advantages and disadvantages. What do you predict will happen? Do you think most pairs will work hard? However, we also worry that you will be tempted to just chat. That would be natural. Here will be the rules. Do you think the rules will work?"
Then you read and respond to the survey results when you try it. Either it's "Most of you think this will work, so let's try" or "Most of you don't think it will work, so let's try it small." But you build belief among students that this is exciting for them, but they need to try hard.
Two plausible ways to start this.
1) One is with just some students while the others are in lecture.
Maybe there are 6 students out of 50 who can work well independently right now. Ie, 3 stronger students and 3 weaker students. You get the 6 kids in the back of the room working very quietly. Maybe you do this while the other 44 are being taught by the teacher in the front. Well of course some of the 44 will look enviously on the 6. Then after a couple days, add 2 more kids -- now 8 are working quietly in the back. Then 10. Each day you try to grow the number of students being tutored.
2) Or you can try everyone at once. Since most of your questions deal with mild misbehavior and loudness issues, you simply set clear expectations. "I want you to whisper. Let's all practice our whisper. Good. Second, I will be walking around and noticing each pair. If you are trying hard, small reward or praise. If not, you will get (small consequence)."
The key for all of this is each day: Set clear goals on your 1 to 10 scale of how hard students will try during peer tutoring; measure whether they do that by walking around and observing (you can even delegate this task to a student); publicly talking to the students about it (Class, today we did a better job of working hard during peer tutoring, thank you; the following pairs really improved over yesterday, thank you); having small rewards (including personal thank yous) and consequences.
Best wishes, Mike