Private messages are written inside the private message field during a donation. You can view a donation in your reporting tools by clicking on the details button on the right side of the transaction line item. The receipt will show if there was a message typed.

A better way to view private messages is be downloading a CSV and then viewing the comment/private message column.

To download a CSV go to your reporting tools from your left menu in your account settings. Make sure you are on "Transactions I Received" tab. Then you will see the button "Download to CSV". Clicking the download button will download a CSV of the donation transaction details.