If someone with a private profile mentions someone who isn't one of their approved followers on a photo or video, the person mentioned won't get a notification in Activity and they won't get a push notification.

If someone with a private profile leaves a comment on a photo that a public Instagrammer uploaded, their comment will be visible and their username will be clickable below that photo.

If someone with a private profile sends a photo/video to a specific group of people with Instagram Direct, only people who follow the private profile will be able to see the comments or mentions associated with that photo/video.