On my last phone, a Droid Razr Maxx, I started using the Verizon Message+ app and really liked it however, the new message icon that previously appeared on the sleep/lock screen ceased appearing. When I switch the default message app back to the factory default, the icon again appears to notify me of a new message without actually waking up or unlocking the phone. I recently got a Turbo 2 and am experiencing the same issue. From the sleep/lock screen, I get notifications from other third-party apps such as Facebook, Twitter, aCalendar and Netgear's Arlo wireless camera app.
Notifications are enabled on the Message+ app and I receive a notification in the taskbar but nothing tells me I have a new message until I swipe down to unlock the screen as is the case with the default messaging app.