At this point in our app, we need to add moveable player racquets and touch input. In Kivy, a widget can react to input by implementing the on_touch_down, the on_touch_move, and the on_touch_up methods. By default, the Widget class implements these methods by just calling the corresponding method on all its child widgets to pass on the event until one of the children returns True.

For the racquets, we essentially just need them to move up and down, and we don't need them to have the player widgets handle the events themselves. So, all we need to do is to implement the on_touch_move function for the PongGame class and have it set the position of left or right based on whether the touch occurs on the left or right side of the screen.

This is the on_touch_move handler:

def on_touch_move(self, touch):
    if touch.x < self.width/3:
        self.player1.center_y = touch.y
    if touch.x > self.width - self.width/3:
        self.player2.center_y = touch.y

Next, we can keep the score for each player in a NumericProperty. The score labels of the PongGame are kept updated by changing the NumericProperty score. The PongPaddle class also implements a bounce_ball method, so that the ball bounces differently based on where it hits the racket.

class PongPaddle(Widget):

    score = NumericProperty(0)

    def bounce_ball(self, ball):
        if self.collide_widget(ball):
            speedup  = 1.1
            offset = 0.02 * Vector(0, ball.center_y-self.center_y)
            ball.velocity =  speedup * (offset - ball.velocity)

And the game is pretty much done! Here is our final code.

main.py:

from kivy.app import App
from kivy.uix.widget import Widget
from kivy.properties import (
    NumericProperty, ReferenceListProperty, ObjectProperty
)
from kivy.vector import Vector
from kivy.clock import Clock


class PongPaddle(Widget):
    score = NumericProperty(0)

    def bounce_ball(self, ball):
        if self.collide_widget(ball):
            vx, vy = ball.velocity
            offset = (ball.center_y - self.center_y) / (self.height / 2)
            bounced = Vector(-1 * vx, vy)
            vel = bounced * 1.1
            ball.velocity = vel.x, vel.y + offset


class PongBall(Widget):
    velocity_x = NumericProperty(0)
    velocity_y = NumericProperty(0)
    velocity = ReferenceListProperty(velocity_x, velocity_y)

    def move(self):
        self.pos = Vector(*self.velocity) + self.pos


class PongGame(Widget):
    ball = ObjectProperty(None)
    player1 = ObjectProperty(None)
    player2 = ObjectProperty(None)

    def serve_ball(self, vel=(4, 0)):
        self.ball.center = self.center
        self.ball.velocity = vel

    def update(self, dt):
        self.ball.move()

        # bounce of paddles
        self.player1.bounce_ball(self.ball)
        self.player2.bounce_ball(self.ball)

        # bounce ball off bottom or top
        if (self.ball.y < self.y) or (self.ball.top > self.top):
            self.ball.velocity_y *= -1

        # went of to a side to score point?
        if self.ball.x < self.x:
            self.player2.score += 1
            self.serve_ball(vel=(4, 0))
        if self.ball.x > self.width:
            self.player1.score += 1
            self.serve_ball(vel=(-4, 0))

    def on_touch_move(self, touch):
        if touch.x < self.width / 3:
            self.player1.center_y = touch.y
        if touch.x > self.width - self.width / 3:
            self.player2.center_y = touch.y


class PongApp(App):
    def build(self):
        game = PongGame()
        game.serve_ball()
        Clock.schedule_interval(game.update, 1.0 / 60.0)
        return game


if __name__ == '__main__':
    PongApp().run()

pong.kv:

#:kivy 1.0.9

<PongBall>:
    size: 50, 50 
    canvas:
        Ellipse:
            pos: self.pos
            size: self.size          

<PongPaddle>:
    size: 25, 200
    canvas:
        Rectangle:
            pos: self.pos
            size: self.size

<PongGame>:
    ball: pong_ball
    player1: player_left
    player2: player_right
    
    canvas:
        Rectangle:
            pos: self.center_x - 5, 0
            size: 10, self.height
    
    Label:
        font_size: 70  
        center_x: root.width / 4
        top: root.top - 50
        text: str(root.player1.score)
        
    Label:
        font_size: 70  
        center_x: root.width * 3 / 4
        top: root.top - 50
        text: str(root.player2.score)
    
    PongBall:
        id: pong_ball
        center: self.parent.center
        
    PongPaddle:
        id: player_left
        x: root.x
        center_y: root.center_y
        
    PongPaddle:
        id: player_right
        x: root.width - self.width
        center_y: root.center_y