BuddyPress allows you to send Public as well as private messages.
For sending a private message head towards your profile >> messages >> Compose.
Here, select a user name in autocomplete “send to” field, type your message and then click on send message button. All done.
For public messages, you can send the same by mentioning the member with “@” sign then your message anywhere within BuddyPress site it can be sitewide activity, group activity or your own profile page. The user will be notified by notification count and via email as well if enabled.