I used to use Invision Community and it had such a feature—On each user’s profile page, there is a tab Activitythat not only lists all activity by the user but also there is a text box at the top that allows you to Write a public message on username's feed...
Though it appears to duplicate the function of private messages, the emphasis was on public because then other users can also see the message and react to or comment on it. It’s like a public yet personal comment to someone that others can join in if they wish, that wouldn’t be relevant as a topic in the forum. I suppose it’s like being able to post on a friend’s Facebook’s personal profile timeline?