I should be more specific. Is there a way to send someone a message that can't be seen by anyone who goes to their profile?
Starting a Conversation IS private. I use my phone so let me make some more detailed steps for you on a PC:
Click someone's name to show the mini-profile, not their Profile Page if that makes sense. Clicking "Start a Conversation" will send them a private message.
Do not go to a whole new page to see their full profile. Click their name once and a tiny popup should come up. Start a Conversation from there, it will be private.
This is not the same thing as Posting to their Profile.