Fix paste and test

src/textual/app.py

@@ -1901,9 +1901,11 @@ class App(Generic[ReturnType], DOMNode):
             else:
                 await self.screen._forward_event(event)
 
-        elif isinstance(event, events.Paste):
+        elif isinstance(event, events.Paste) and not event.is_forwarded:
             if self.focused is not None:
                 await self.focused._forward_event(event)
+            else:
+                await self.screen._forward_event(event)
         else:
             await super().on_event(event)
 

tests/test_paste.py

@@ -0,0 +1,18 @@
+from textual.app import App
+from textual import events
+
+
+async def test_paste_app():
+    paste_events = []
+
+    class PasteApp(App):
+        def on_paste(self, event):
+            paste_events.append(event)
+
+    app = PasteApp()
+    async with app.run_test() as pilot:
+        await app.post_message(events.Paste(sender=app, text="Hello"))
+        await pilot.pause(0)
+
+    assert len(paste_events) == 1
+    assert paste_events[0].text == "Hello"