When receiving a message, you will get a notification, as long as these are enabled. Furthermore, when entering the game, the chat icon will display a number representing the number of conversations with new unread messages.
Also, when entering the conversations list, a new number will be displayed next to each user’s name indicating the number of unread messages received from them. A checkmark will appear below the message once it has been sent. When the message reaches the other player, a double checkmark will appear.