There is no way to delete a direct message on both sides. The person you sent the message to will still have access to it in their inbox unless they choose to delete it themselves.
If you want to delete a message permanently, you can do so by holding down the message and clicking ‘Unsend’. This will remove the message from your chat history, as well as the other person’s chat history.
However, it is important to note that unsending a message does not guarantee that the other person will not be able to see it. It is possible for them to take a screenshot of the message before it is deleted, or to otherwise retrieve the information from their device.
How do you delete an entire conversation on Instagram on both sides?

It’s not possible to delete an entire conversation on Instagram from both sides. The app only allows users to unsend messages that they’ve sent, or to delete the entire conversation.
To delete a message in a chat, hold down the message and then click “Unsend.” This will permanently delete the message from your chat.
Does deleting an Instagram conversation delete it for the other person?

The person you are messaging will still be able to see the conversation unless they delete it from their end as well. So if you’re trying to get rid of evidence, you’ll both have to delete the entire conversation.