diff --git a/asgiref/local.py b/asgiref/local.py
index a8b9459b..784c5709 100644
--- a/asgiref/local.py
+++ b/asgiref/local.py
@@ -24,9 +24,8 @@ def __setattr__(self, key: str, value: Any) -> None:
         if key == "_data":
             return super().__setattr__(key, value)
 
-        storage_object = self._data.get({})
-        storage_object[key] = value
-        self._data.set(storage_object)
+        # Update a copy of the existing storage.
+        self._data.set({**self._data.get({}), key: value})
 
     def __delattr__(self, key: str) -> None:
         storage_object = self._data.get({})
diff --git a/tests/test_local.py b/tests/test_local.py
index d50cba21..32cd6cd3 100644
--- a/tests/test_local.py
+++ b/tests/test_local.py
@@ -1,9 +1,18 @@
+# type: ignore
+
 import asyncio
 import gc
 import threading
+import unittest
+from collections.abc import Callable
+from contextvars import ContextVar
+from threading import Thread
+from typing import Any
+from unittest import TestCase
 
 import pytest
 
+import asgiref
 from asgiref.local import Local
 from asgiref.sync import async_to_sync, sync_to_async
 
@@ -338,3 +347,124 @@ async def async_function():
     # inner value was set inside a new async context, meaning that
     # we do not see it, as context vars don't propagate up the stack
     assert not hasattr(test_local_not_tc, "test_value")
+
+
+from packaging.version import Version
+
+is_new_asgiref = Version(asgiref.__version__) >= Version("3.7")
+
+
+def run_in_thread(func: Callable[..., Any], *args: Any, **kwargs: Any) -> Any:
+    """Run a callable in a thread."""
+    result: Any = None
+
+    def _thread_main() -> None:
+        nonlocal result
+        result = func(*args, **kwargs)
+
+    thread = Thread(target=_thread_main)
+    thread.start()
+    thread.join()
+
+    return result
+
+
+def run_in_task(func: Callable[..., Any], *args: Any, **kwargs: Any) -> Any:
+    """Run a callable in an asyncio task."""
+
+    async def _task_main():
+        return func(*args, **kwargs)
+
+    async def _async_main():
+        return await asyncio.create_task(
+            _task_main(),
+        )
+
+    return asyncio.run(_async_main())
+
+
+local = Local()
+cvar: ContextVar[int] = ContextVar("cvar")
+
+
+class Test(TestCase):
+    """Test Local visibility behaviour."""
+
+    def setUp(self):
+        setattr(local, "value", 0)
+        cvar.set(0)
+
+    def test_visibility_thread_asgiref(self) -> None:
+        """Check visibility with subthreads."""
+        self.assertEqual(local.value, 0)
+
+        def _test():
+            # Local() is cleared when changing thread
+            self.assertFalse(hasattr(local, "value"))
+            setattr(local, "value", 1)
+            self.assertEqual(local.value, 1)
+
+        run_in_thread(_test)
+
+        self.assertEqual(local.value, 0)
+
+    def test_visibility_thread_contextvar(self) -> None:
+        """Check visibility with subthreads."""
+        self.assertEqual(cvar.get(), 0)
+
+        def _test():
+            # ContextVar is cleared when changing thread
+            with self.assertRaises(LookupError):
+                cvar.get()
+            cvar.set(1)
+            self.assertEqual(cvar.get(), 1)
+
+        run_in_thread(_test)
+
+        self.assertEqual(cvar.get(), 0)
+
+    @unittest.skipIf(is_new_asgiref, "test for old asgiref")
+    def test_visibility_task_asgiref_pre_37(self) -> None:
+        """Check visibility with asyncio tasks."""
+        self.assertEqual(local.value, 0)
+
+        def _test():
+            # Local is cleared on pre-3.7 when changing task
+            self.assertFalse(hasattr(local, "value"))
+            setattr(local, "value", 1)
+            self.assertEqual(local.value, 1)
+
+        run_in_task(_test)
+
+        self.assertEqual(local.value, 0)
+
+    @unittest.skipIf(not is_new_asgiref, "test for new asgiref")
+    def test_visibility_task_asgiref_post_37(self) -> None:
+        """Check visibility with asyncio tasks."""
+        self.assertEqual(local.value, 0)
+
+        def _test():
+            # Local is inherited on 3.7+ when changing task
+            self.assertEqual(local.value, 0)
+            local.value = 1
+            self.assertEqual(local.value, 1)
+
+        run_in_task(_test)
+
+        # Changes leak to the caller, and probably should not
+        self.assertEqual(local.value, 0)
+        self.assertEqual(getattr(local, "value"), 0)
+
+    def test_visibility_task_contextvar(self) -> None:
+        """Check visibility with subthreads."""
+        self.assertEqual(cvar.get(), 0)
+
+        def _test():
+            # ContextVar is inherited when changing task
+            self.assertEqual(cvar.get(), 0)
+            cvar.set(1)
+            self.assertEqual(cvar.get(), 1)
+
+        run_in_task(_test)
+
+        self.assertEqual(cvar.get(), 0)