Sending private messages cannot be done through the Facebook app on your iPhone. You will need to download and install the Facebook Messenger app in order to read or send a private message.
To install the app, open Facebook on your iPhone. Then tap the Messages icon in the bottom toolbar.
Next, tap the blue button which says Get Messenger. This will bring you to the App Store.
Tap the Install button, and Messenger will start downloading to your iPhone.
