Twitter lets you start a private conversations with your audience when interacting through Tweets. All you have to do is to add a Direct Message link to your tweets to allow your audience share certain information such as phone numbers, contact information, or personal details, through a Direct Message. So, when the user clicks on the link, they'll be automatically be directed from their public timeline to a private Direct Message along with your Tweet, to give you a context of the interaction. 

Here's how you can add Direct Message links with your Tweets: