TestClass-private-class.java

public class TestClass {
    public static void main(String... args) {
        new Inner();
    }

    // Private class, package-level constructor. 
    private static class Inner {
        Inner() {
            System.out.println("Hello, world.");
        }
    }
}

TestClass-private-class.javap

Compiled from "TestClass.java"
class TestClass$Inner {
  TestClass$Inner();
    Code:
       0: aload_0
       1: invokespecial #1                  // Method java/lang/Object."<init>":()V
       4: getstatic     #2                  // Field java/lang/System.out:Ljava/io/PrintStream;
       7: ldc           #3                  // String Hello, world.
       9: invokevirtual #4                  // Method java/io/PrintStream.println:(Ljava/lang/String;)V
      12: return
}
Compiled from "TestClass.java"
public class TestClass {
  public TestClass();
    Code:
       0: aload_0
       1: invokespecial #1                  // Method java/lang/Object."<init>":()V
       4: return

  public static void main(java.lang.String...);
    Code:
       0: new           #2                  // class TestClass$Inner
       3: dup
       4: invokespecial #3                  // Method TestClass$Inner."<init>":()V
       7: pop
       8: return
}

TestClass-private-constructor.java

public class TestClass {
    public static void main(String... args) {
        new Inner();
    }

    static class Inner {
        private Inner() {
            System.out.println("Hello, world.");
        }
    }
}

TestClass-private-constructor.javap

Compiled from "TestClass.java"
class TestClass$1 {
}
Compiled from "TestClass.java"
class TestClass$Inner {
  private TestClass$Inner();
    Code:
       0: aload_0
       1: invokespecial #2                  // Method java/lang/Object."<init>":()V
       4: getstatic     #3                  // Field java/lang/System.out:Ljava/io/PrintStream;
       7: ldc           #4                  // String Hello, world.
       9: invokevirtual #5                  // Method java/io/PrintStream.println:(Ljava/lang/String;)V
      12: return

  TestClass$Inner(TestClass$1);
    Code:
       0: aload_0
       1: invokespecial #1                  // Method "<init>":()V
       4: return
}
Compiled from "TestClass.java"
public class TestClass {
  public TestClass();
    Code:
       0: aload_0
       1: invokespecial #1                  // Method java/lang/Object."<init>":()V
       4: return

  public static void main(java.lang.String...);
    Code:
       0: new           #2                  // class TestClass$Inner
       3: dup
       4: aconst_null
       5: invokespecial #3                  // Method TestClass$Inner."<init>":(LTestClass$1;)V
       8: pop
       9: return
}

Private inner classes don't add additional overhead in terms of methods or class files, whereas private constructors infamously do.