Yes, it's possible to start a conversation with a contact, but only if they have been in touch with you before.
You can start a new conversation by looking them up in your "Contacts" tab. Select the contact from the list, and a button will appear to initiate a new conversation:
Another way of engaging an existing contact is by resuming a finished conversation or starting a new one. You can find your completed conversations in your Archive:
A conversation in status "Open" or "Pending" can always be continued by sending a message.