It depends on their Direct Message (DM) settings. By default, Twitter allows DMs only from mutual followers. This means that if a private account does not follow you back, you will not be able to send them a message.

However, if the private account has enabled the option to receive DMs from "Everyone," then you can send them a message even if they do not follow you. But this setting is optional, and many private accounts keep it disabled for added privacy.

If you try to DM a private account that does not allow messages from non-followers, you will see a notification stating, "You can’t message this account unless they follow you."

To learn more about private Twitter accounts and how to interact with them, you can read private tweets.