To send a photo in private chat, you will first need to click on the 3 lines in the upper right corner of the message screen.
Once that is clicked, a pop up will come from the bottom of the screen. You will want to select, Send Picture!
Once Send Picture is selected, a second pop up from the bottom of the screen will appear. Select either Take Photo or Choose Existing Photo.
Take Photo will take you to your devices camera.
Choose Existing Photo will take you to your devices photo gallery.
Select, and it will now send that user your private photo!