Not sure about wired headsets, but with Bluetooth, it depends on the product. If you are using a Samsung Gear Circle or Level device, there are apps for them you can install that allow you to control what happens when a notification is incoming. you can choose from basic details, which will just say who the message is from, or you can opt for more detailed notifications which read out the message and who it is from.
Going through phone settings, there isn't anything that I can find that will allow a wired connection to read out notifications
