You'll see the green dot in the Direct inbox and in your friends list when you share a post from the feed. Dots will be displayed for friends who follow you or people you have previously spoken to in Direct.
If desired, you can hide your own status and turn off the ability to see when your friends are active in the settings section of the Instagram app.
Instagram says this new update is meant to make it easier to connect with friends over direct messages for more realtime conversations. The dots are rolling out to Instagram users today.
Instagram can be downloaded from the App Store for free. [Direct Link]