Twitter calls its chat service Direct Message (DM). By default you can only send someone a DM if the person follows you. Back on the early years of Twitter, you had to ask the person to follow you back in order to have a private conversation.
For certain Twitter account, it could be useful to allow anyone to send you a Direct Message. For example you might want to setup a support Twitter account to receive messages from anyone in order to attract all potential customers.
To setup to receive DM from anyone on the web, login to your account on twitter.com. Go to account Settings > Security and privacy. Scroll down to locate “Receive Direct Messages from anyone”, check the tick box and save the changes.
To setup on your iOS devices, run the official Twitter app. Select the Me tab and tap on the gear icon. Select Settings from the menu that pops up, and then select the Twitter account you’d like to setup. Under Privacy section, locate and switch on “Receive Direct Messages from anyone”.