Like in Facebook, when you click ‘Message’ button present on the profile page, a pop-up appears from where you can message to the person without leaving his/her profile. Whereas by default in BuddyPress as soon you click ‘Message’ button you get redirected to your message compose box for creating and sending message to the person.
This tutorial explains you how you can easily replaces the functionality for Public Message and Private Message buttons on profile pages so that you can send messages without leaving the profile page.
BP Profile Message UX Free plugin lets you replace the functionality for those message buttons. Using this plugin you can make things go fast, you can stay on a profile page and send private or public messages to member.