Direct Messaging
When you and another user have followed each other, you can then use the direct messaging facility. This enables you to message each other, privately, without all your respective followers seeing it. You can change your privacy settings, too, to accept DMs from anyone, but this is not recommended.
Unwelcome DMs
You will get DMs which are unwelcome anyway even from some followers, so opening this up to anyone, on Twitter, is not very wise. DMs also allow much longer messages than a standard tweet. However, DMs have some issues attached. Firstly, a Twitter follower is not the same as a Facebook friend, so they may not be as keen to message, one to one.
They signed up to read your public content, not necessarily for a deeper relationship. In addition, scam accounts have added to the problem. They follow you to get a follow back and soon you are receiving messages, selling you love, Bitcoin, share options, cancer sob stories involving sending donations, etc. For this reason many people are not comfortable with being messaged, directly. You will see in some bios it will say “No DMs” or similar.
The suggestion is to use these sparingly and, usually, to ask, via a public reply to a tweet, whether the individual is comfortable to receive these and explain why one don’t want to share the message, to a wider public.
It is recommended that you are cautious in using DMs. One or two peopleeven send a “thank you” DM, to people who follow them back, but this is about as far as it would be wise to go. The best is to seek permission first.