Does the C# compiler treat a lambda expression as a public or private method?

It depends. With the current version of Visual Studio, the methods that implement lambdas are never public, but they’re not always private. A simple program to test some versions of lambdas:

public class Program
{
    public static void Main()
    {
        var program = new Program();
        Try("A", program.A);
        Try("B", program.B);
        Try("C", program.C);
        Console.ReadKey();
    }

    private static void Try(string name, Func<Action> generator)
    {
        var mi = generator().Method;
        Console.WriteLine($"{name}: DeclaringType={mi.DeclaringType}, Attributes={mi.Attributes}");
    }

    private Action A() => () => { };
    private Action B() => () => { ToString(); };
    private Action C()
    {
        var c = 1;
        return () => c.ToString();
    }
}

prints

A: DeclaringType=Scratch.Program+<>c, Attributes=PrivateScope, Assembly, HideBySig
B: DeclaringType=Scratch.Program, Attributes=PrivateScope, Private, HideBySig
C: DeclaringType=Scratch.Program+<>c__DisplayClass4_0, Attributes=PrivateScope, Assembly, HideBySig

A‘s lambda doesn’t have any captures. It’s created as an internal method of an empty closure class.

B‘s lambda captures this. It’s created as a private method of the containing class.

C‘s lambda captures c. It’s created as an internal method of a non-empty closure class.

All of this is undocumented and has changed in the past, so it would be good to avoid relying on it. What matters is that when you call the anonymous method, it behaves as specified. If you need anything more than that, you shouldn’t be using anonymous methods. Depending on what you’re after, you might either still be able to use lambdas, but with expression trees, or you might need to create regular named methods instead.