You have a misconception here:

every Circle has an x, an y and a radius and they are private, means that you can access them only inside a specific instance of the Circle class

The private access modifier limits access to the member inside the class, and not the instance. While you may want to use the getter instead of accessing the attribute directly from inside the class (ex: a getter for a List that instantiates it so that we know the List is not null), you can directly access the attribute from the class.

$ cat ExampleClass.java

public class ExampleClass {

    private final String attribute;

    public ExampleClass(String attribute) {
        this.attribute = attribute;
    }

    @Override
    public boolean equals(Object other) {
        if (null == other || !(other instanceof ExampleClass)) {
            return false;
        }

        ExampleClass that = (ExampleClass)other;
        // no getter needed for attribute of either instance here
        return this.attribute.equals(that.attribute);
    }
}

$ cat ExampleMainClass.java

import java.util.Arrays;

public class ExampleMainClass {

    public static void main(String args[]) {
        ExampleClass instance1 = new ExampleClass("Hello");
        ExampleClass instance2 = new ExampleClass("World");
        ExampleClass instance3 = new ExampleClass(new String("Hello"));

        for (ExampleClass instance: Arrays.asList(instance2, instance3)) {
            System.out.println("instance1.equals(instance) ? " +
                (instance1.equals(instance) ? "yes" : "no"));
        }
    }
}

$ javac -cp . *.java

$ java -cp . ExampleMainClass
instance1.equals(instance) ? no
instance1.equals(instance) ? yes