diff --git a/swh/storage/tests/test_replay.py b/swh/storage/tests/test_replay.py
--- a/swh/storage/tests/test_replay.py
+++ b/swh/storage/tests/test_replay.py
@@ -269,8 +269,9 @@
     ]
 
 
+@pytest.mark.parametrize("privileged", [True, False])
 def test_storage_play_anonymized(
-    kafka_prefix: str, kafka_consumer_group: str, kafka_server: str
+    kafka_prefix: str, kafka_consumer_group: str, kafka_server: str, privileged: bool,
 ):
     """Optimal replayer scenario.
 
@@ -278,6 +279,7 @@
     - writes objects to the topic
     - replayer consumes objects from the topic and replay them
 
+    This tests the behavior with both a privileged and non-privileged replayer
     """
     writer_config = {
         "cls": "kafka",
@@ -300,37 +302,24 @@
         method(objs)
         nb_sent += len(objs)
 
-    # Fill a destination storage from Kafka **using anonymized topics**
+    # Fill a destination storage from Kafka, potentially using privileged topics
     dst_storage = get_storage(cls="memory")
     replayer = JournalClient(
         brokers=kafka_server,
         group_id=kafka_consumer_group,
         prefix=kafka_prefix,
         stop_after_objects=nb_sent,
-        privileged=False,
+        privileged=privileged,
     )
     worker_fn = functools.partial(process_replay_objects, storage=dst_storage)
 
     nb_inserted = replayer.process(worker_fn)
     replayer.consumer.commit()
-    assert nb_sent == nb_inserted
-    check_replayed(storage, dst_storage, expected_anonymized=True)
-
-    # Fill a destination storage from Kafka **with stock (non-anonymized) topics**
-    dst_storage = get_storage(cls="memory")
-    replayer = JournalClient(
-        brokers=kafka_server,
-        group_id=f"{kafka_consumer_group}-2",
-        prefix=kafka_prefix,
-        stop_after_objects=nb_sent,
-        privileged=True,
-    )
-    worker_fn = functools.partial(process_replay_objects, storage=dst_storage)
 
-    nb_inserted = replayer.process(worker_fn)
-    replayer.consumer.commit()
     assert nb_sent == nb_inserted
-    check_replayed(storage, dst_storage, expected_anonymized=False)
+    # Check the contents of the destination storage, and whether the anonymization was
+    # properly used
+    check_replayed(storage, dst_storage, expected_anonymized=not privileged)
 
 
 def check_replayed(src, dst, expected_anonymized=False):