passingdatabetweenscreens.kv

#:kivy 1.5.1
#:import kivy kivy

<ScreenOne>:
    GridLayout:
        size: root.size
        rows: 3

        Label:
            text: 'Screen One'

        TextInput:
            id: text_screen_one

        Button:
            text: 'Go to Screen Two'
            on_press: app.thestring = text_screen_one.text; root.manager.current = 'screentwo'


<ScreenTwo>:
    GridLayout:
        size: root.size
        rows: 3

        Label:
            text: 'Screen Two'

        TextInput:
            id: text_screen_two
            text: app.thestring
            focus: True

        Button:
            text: 'Go to Screen One'
            on_press: root.manager.current = 'screenone'

PassingDataBetweenScreens.py

from kivy.app import App
from kivy.properties import StringProperty
from kivy.uix.screenmanager import ScreenManager, Screen

class ScreenOne(Screen):
    pass

class ScreenTwo(Screen):
    pass

class PassingDataBetweenScreensApp(App):

    thestring = StringProperty('')

    def build(self):

        sm = ScreenManager()
        sm.add_widget(ScreenOne(name='screenone'))
        sm.add_widget(ScreenTwo(name='screentwo'))

        return sm

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