Im a new owner to a LG G2 and I have noticed that when I get a text, the phone will not show it until I unlock the screen.
The notification light will sometimes work, but not all the time.
You can use a 3rd party app like Dynamic Notifications or another in the similar app list from that app in Play Store. You can try to use Verizon Messages to get your SMS and it will light up your screen and unlock it. Or, and this is the best battery saving option, as well as security...wait for Lollipop 5.0 in a couple months. I have tried all the apps that do what you want. They are mostly based on the Motorola notification system, but Moto uses a different screen tech than LG, and using that on ours will kill a battery quick, and they aren't that reliable.
Is there a reason this happens?
My old phone popped the messages up?
It isn't an app, code, or API written into the LG skin of the Android OS.