Twitter just announced today a new feature that allows users to opt-in to receiving Direct Messages (DMs) from anybody including people that you don't follow. In order to be able to receive DM, you would need to enable the feature from the settings menu. Here's a guide on how to enable this feature on your Twitter app.
1. You need to acces the settings menu, which can be accessed by tapping on the three dot menu and choosing Settings from the Options.
2. Next, select the account that you want to enable this feature for.
3. Scroll down and select the Privacy option.
4. Check the option to Receive DM from anyone.