Multiplayer game with java

I wrote little game like chess on Netbeans/Java with Server-Client structure. Two players are playing with each other if they are at same wifi or same lan. Now I want to make it " play with different networks ". How can I do that ? I have 2 idea but I need advice. Firstly I will write my Client-Server connection code.

Client.Start("127.0.0.1", 2000);
 public static void Start(String ip, int port) {
        try {
            // Client Socket object
            Client.socket = new Socket(ip, port);
            Client.Display("Connected to server");
            // input stream
            Client.sInput = new ObjectInputStream(Client.socket.getInputStream());
            // output stream
            Client.sOutput = new ObjectOutputStream(Client.socket.getOutputStream());
            Client.listenMe = new Listen();
            Client.listenMe.start();

            Message msg = new Message(Message.Message_Type.Name);
            msg.content = Game.ThisGame.txt_name.getText();
            Client.Send(msg);
        } catch (IOException ex) {
            Logger.getLogger(Client.class.getName()).log(Level.SEVERE, null, ex);
        }
    }

Main Purpose : Players on different network can play this.

my First idea is - rent a Server . and Use that server like my clients's server. But Idk , Servers can run Java code ?

Second idea is implementing my netbeans code for connect to online database and coding my moves , when client took enemy's moves on database , program will encode it and shows moves on my board. Thanks for advices,ideas