Sending messages between two JPanel objects

For mKorbel (and the original poster):
What I'm recommending is looser coupling, that the one JPanel has no knowledge of the other JPanel and that all connections are done through a control of some sort. For instance, to borrow some of your code:

CopyTextNorthPanel2.java

import java.awt.*;
import javax.swing.*;

public class CopyTextNorthPanel2 extends JPanel {

   private static final long serialVersionUID = 1L;
   public JTextField northField;

   public CopyTextNorthPanel2() {
      northField = new JTextField("Welcome World");
      northField.setFont(new Font("Serif", Font.BOLD, 20));
      northField.setPreferredSize(new Dimension(300, 25));
      add(northField);
   }

   public String getNorthFieldText() {
      return northField.getText();
   }
}

CopyTextSouthPanel2.java

import java.awt.event.*;
import javax.swing.*;

public class CopyTextSouthPanel2 extends JPanel {

   private static final long serialVersionUID = 1L;
   private JTextField firstText = new JTextField("Desired TextField");
   private JButton copyButton = new JButton("Copy text from JTextFields");
   private CopyTextControl2 control;

   public CopyTextSouthPanel2() {
      copyButton.addActionListener(new ActionListener() {
         public void actionPerformed(ActionEvent e) {
            if (control != null) {
               control.copyAction();
            }
         }
      });

      add(firstText);
      add(copyButton);
   }

   public void setControl(CopyTextControl2 control) {
      this.control = control;
   }

   public void setFirstText(String text) {
      firstText.setText(text);
   }
}

CopyTextControl2.java

public class CopyTextControl2 {
   private CopyTextNorthPanel2 northPanel;
   private CopyTextSouthPanel2 southPanel;

   public void copyAction() {
      if (northPanel != null && southPanel != null) {
         southPanel.setFirstText(northPanel.getNorthFieldText());
      }
   }

   public void setNorthPanel(CopyTextNorthPanel2 northPanel) {
      this.northPanel = northPanel;
   }

   public void setSouthPanel(CopyTextSouthPanel2 southPanel) {
      this.southPanel = southPanel;
   }

}

CopyText2.java

import java.awt.*;
import javax.swing.*;

public class CopyText2 {

   private static void createAndShowUI() {
      CopyTextNorthPanel2 northPanel = new CopyTextNorthPanel2();
      CopyTextSouthPanel2 southPanel = new CopyTextSouthPanel2();
      CopyTextControl2 control = new CopyTextControl2();

      southPanel.setControl(control);
      control.setNorthPanel(northPanel);
      control.setSouthPanel(southPanel);

      JPanel mainPanel = new JPanel(new BorderLayout());
      mainPanel.add(northPanel, BorderLayout.NORTH);
      mainPanel.add(Box.createRigidArea(new Dimension(100, 100)), BorderLayout.CENTER);
      mainPanel.add(southPanel, BorderLayout.SOUTH);

      JFrame frame = new JFrame("Copy Text");
      frame.getContentPane().add(mainPanel);
      frame.setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
      frame.pack();
      frame.setLocationRelativeTo(null);
      frame.setVisible(true);
   }

   public static void main(String[] args) {
      java.awt.EventQueue.invokeLater(new Runnable() {
         public void run() {
            createAndShowUI();
         }
      });
   }
}

You could create a custom event, and attach one or more listeners to it.

The proper way to implement is to have the Button ActionListener fire the event and then have your two panels be listeners for that event.

at the top of your class

public MyClass implements ActionListener {

...
myButton.addActionListener(this);
...

public void actionPerformed(ActionEvent e) {
      //for example if you have more than one events that you need to handle
      if(e.getSource().equals(myButton) {
           //update your do some work on you jpanels
      }
}

But really, I think it is time to start thinking about design patterns. What you are describing is perfect candidate for the observer pattern and possibly the command pattern