In Java, when should we use private instance methods in interfaces?

Interfaces are used to define an object's behaviour. This means all of the interface's methods are exposed. When using default methods, we can provide standard implementations of the defined methods, offering code reuse across class boundaries.

In some cases, functionality is required (perhaps just for code reuse in different default methods) but should not be exposed because it would pollute class'/object's namespaces. This is where private default methods come in handy. Examples private default methods could be factories, validations or default state handling.

package com.company;

import java.util.List;
import java.util.function.Predicate;
import java.util.stream.Collectors;

public class Main {

  public static void main(final String[] args) {
    var messages =
        List.of(
            MessageQueue.newSubject("Message 1"),
            MessageQueue.newTopic("Message 2"),
            MessageQueue.newTopic("Message 3"));
    final MessageQueueAdapter1 queue1 = () -> messages;
    inspectQueue(queue1);
    final MessageQueueAdapter2 queue2 = () -> messages;
    inspectQueue(queue2);
  }

  private static void inspectQueue(final MessageQueue queue) {
    final List<Message> messagesWithSubject = queue.getMessagesWithSubject();
    assert messagesWithSubject.size() == 1 : "expected one message with 'Subject'";
    final List<Message> messagesWithTopic = queue.getMessagesWithTopic();
    assert messagesWithTopic.size() == 2 : "expected two message with 'Topic'";
    assert !queue.getMessages().isEmpty() && 3 == queue.getMessages().size()
        : "expected three messages in total";
  }

  @FunctionalInterface
  interface Message {
    private static boolean isPrefixedBy(final String message, final String prefix) {
      return message != null && !message.isEmpty() && message.startsWith(prefix);
    }

    default boolean hasSubject() {
      return isPrefixedBy(this.getMessage(), MessageQueue.PREFIX_SUBJECT);
    }

    default boolean hasTopic() {
      return isPrefixedBy(this.getMessage(), MessageQueue.PREFIX_TOPIC);
    }

    String getMessage();
  }

  interface MessageQueue {
    String PREFIX_SUBJECT = "Subject: ";

    String PREFIX_TOPIC = "Topic: ";

    private static Message newMessage(final String message) {
      return () -> message;
    }

    static Message newSubject(final String message) {
      return newMessage(PREFIX_SUBJECT + message);
    }

    static Message newTopic(final String message) {
      return newMessage(PREFIX_TOPIC + message);
    }

    List<Message> getMessages();

    List<Message> getMessagesWithSubject();

    List<Message> getMessagesWithTopic();
  }

  @FunctionalInterface
  interface MessageQueueAdapter1 extends MessageQueue {
    private static List<Message> filterBy(
        final List<Message> messages, final Predicate<Message> predicate) {
      return messages.stream().filter(predicate).collect(Collectors.toList());
    }

    /** {@inheritDoc} */
    @Override
    default List<Message> getMessagesWithSubject() {
      return filterBy(this.getMessages(), Message::hasSubject);
    }

    /** {@inheritDoc} */
    @Override
    default List<Message> getMessagesWithTopic() {
      return filterBy(this.getMessages(), Message::hasTopic);
    }
  }

  @FunctionalInterface
  interface MessageQueueAdapter2 extends MessageQueue {
    private List<Message> filterBy(final Predicate<Message> predicate) {
      return this.getMessages().stream().filter(predicate).collect(Collectors.toList());
    }

    /** {@inheritDoc} */
    @Override
    default List<Message> getMessagesWithSubject() {
      return filterBy(Message::hasSubject);
    }

    /** {@inheritDoc} */
    @Override
    default List<Message> getMessagesWithTopic() {
      return filterBy(Message::hasTopic);
    }
  }
}

OK, another attempt at actually answering OP's questions. When you need to call another non-static method on the interface from a private method, the private method cannot be static. For example, there would be a compilation error if the private method below was static:

public interface InterfaceWithMethods {
    public default void doSomething() {
        doSomethingCommon();
    }

    public default void doSomethingElse() {
        doSomethingCommon();
    }

    public void actuallyDoSomething();

    private void doSomethingCommon() {
        System.out.println("Do something first.");
        actuallyDoSomething();
    }
}