You can send private messages to your contacts or when sending connection and meeting requests.
To send a new message, simply go to their profile and write a message in the chat bar on the right. You can access a contacts profile by going to "my contacts."
During an event, you can search for an attendee under the "attendee" tab, select a profile, and send them a message.
To send a message to someone you have already chatted with, click on the on your navigation bar to open your message board. There you will see all messages that you have sent and received from people.