The messaging tool can be configured for the entire instance from Admin Panel > Tools and Utilities > Messages. 

Upon accessing, you will find several options ranging from not allowing students to use messaging functions, to allowing communication with any user in their courses.

You can also configure messaging differently in courses and organizations.

In General Message Settings you can completely block students from using the tool by checking the box.

In Ultra Message Settings, you can use these options to allow replies by email to personal and/or student messages received in email, thus moving the conversation outside of Blackboard.

Finally, the options in Ultra Course/Organization Message Settings will allow you to control whom students can send messages to: everyone in the course, only staff and members of their groups, only staff, or simply reply to messages without the ability to start a new conversation.

Additionally, you can force a copy of each message to be sent by email. In some cases, it is useful to allow this setting to be modified from within the course itself. For this, there is an option in Override Course Level. In this case, users with the appropriate privilege could edit these messaging options (who students can write to) from Course Settings.

If you disable messaging from Tools or from a node, all generated messages will go directly to students' email, without leaving any trace in Blackboard.