Currently, there is no easy way to monitor a child's text messages on Verizon, even if the child is on your account. For instance, there is no online monitoring system where you can read messages. What you can do is either ask to see the child's text messages on their phone, or buy them a SmartPhone or BlackBerry and purchase an SMS Text Messaging monitoring system.
Purchase a mobile monitoring system for your child's cell phone directly from the mobile monitoring provider, not from Verizon, which does not have this service. These systems have various features, such as text message monitoring and only allowing calls from certain numbers, that you can customize. Each plan has a monthly fee and each is only compatible with certain types of mobile phones.
Establish parental notification controls on the mobile monitoring system. This tells the system where to email information and under which conditions you wish to be notified.
Take an in-depth online tutorial of the system you chose to purchase, so you'll know how to use it appropriately.