diff --git a/haystack/errors.py b/haystack/errors.py
index 9fceb61482..b6fb4a5248 100644
--- a/haystack/errors.py
+++ b/haystack/errors.py
@@ -12,10 +12,15 @@ class HaystackError(Exception):
     This error wraps its source transparently in such a way that its attributes
     can be accessed directly: for example, if the original error has a `message` attribute,
     `HaystackError.message` will exist and have the expected content.
+    If send_message_in_event is set to True (default), the message will be sent as part of a telemetry event reporting the error.
+    The messages of errors that might contain user-specific information will not be sent, e.g., DocumentStoreError or OpenAIError.
     """
 
-    def __init__(self, message: Optional[str] = None, docs_link: Optional[str] = None):
-        send_custom_event(event=f"{type(self).__name__} raised")
+    def __init__(
+        self, message: Optional[str] = None, docs_link: Optional[str] = None, send_message_in_event: bool = True
+    ):
+        payload = {"message": message} if send_message_in_event else {}
+        send_custom_event(event=f"{type(self).__name__} raised", payload=payload)
         super().__init__()
         if message:
             self.message = message
@@ -72,8 +77,8 @@ def __init__(
 class DocumentStoreError(HaystackError):
     """Exception for issues that occur in a document store"""
 
-    def __init__(self, message: Optional[str] = None):
-        super().__init__(message=message)
+    def __init__(self, message: Optional[str] = None, send_message_in_event: bool = False):
+        super().__init__(message=message, send_message_in_event=send_message_in_event)
 
 
 class FilterError(DocumentStoreError):
@@ -100,8 +105,8 @@ def __init__(self, message: Optional[str] = None):
 class NodeError(HaystackError):
     """Exception for issues that occur in a node"""
 
-    def __init__(self, message: Optional[str] = None):
-        super().__init__(message=message)
+    def __init__(self, message: Optional[str] = None, send_message_in_event: bool = True):
+        super().__init__(message=message, send_message_in_event=send_message_in_event)
 
 
 class AudioNodeError(NodeError):
@@ -114,8 +119,10 @@ def __init__(self, message: Optional[str] = None):
 class OpenAIError(NodeError):
     """Exception for issues that occur in the OpenAI APIs"""
 
-    def __init__(self, message: Optional[str] = None, status_code: Optional[int] = None):
-        super().__init__(message=message)
+    def __init__(
+        self, message: Optional[str] = None, status_code: Optional[int] = None, send_message_in_event: bool = False
+    ):
+        super().__init__(message=message, send_message_in_event=send_message_in_event)
         self.status_code = status_code
 
 
@@ -126,13 +133,15 @@ class OpenAIRateLimitError(OpenAIError):
     See https://help.openai.com/en/articles/5955598-is-api-usage-subject-to-any-rate-limits
     """
 
-    def __init__(self, message: Optional[str] = None):
-        super().__init__(message=message, status_code=429)
+    def __init__(self, message: Optional[str] = None, send_message_in_event: bool = False):
+        super().__init__(message=message, status_code=429, send_message_in_event=send_message_in_event)
 
 
 class CohereError(NodeError):
     """Exception for issues that occur in the Cohere APIs"""
 
-    def __init__(self, message: Optional[str] = None, status_code: Optional[int] = None):
-        super().__init__(message=message)
+    def __init__(
+        self, message: Optional[str] = None, status_code: Optional[int] = None, send_message_in_event: bool = False
+    ):
+        super().__init__(message=message, send_message_in_event=send_message_in_event)
         self.status_code = status_code
diff --git a/haystack/telemetry.py b/haystack/telemetry.py
index d3abce6416..faf94fb0af 100644
--- a/haystack/telemetry.py
+++ b/haystack/telemetry.py
@@ -294,6 +294,7 @@ class NonPrivateParameters:
         "uptime",
         "run_total",
         "run_total_window",
+        "message",
     ]
 
     @classmethod