If you mean monitoring the content, no - you need to view the texts on the phone itself. You can see who (what numbers) are being texted, unless it's an iPhone and they are using iMessage, or a smartphone and a texting app is being used.
Depending on the age of the child, I would (and I did with my own kids) make it a condition of having the phone that I have access to the phone at any time to view who and what is being texted - I'm paying the bill, what is transmitted on the phone is my responsibility. Once they got a job, and could pay for their own phone, they could choose to be on our family plan, subject to review at any time, or pay for their own if they felt they needed privacy.
What type of phone does your child have?
If it is an android based phone, you can install Verizon Messages from the Play Store. This app can sync the texts which are sent to/from the phone with a website you can view on a computer. You would have to be logged into the website as the MyVerizon account associated with the phone, though.
As a result of being synced, if a message is deleted on the phone, it will also be deleted on the website.
yes you can, first get family share plan, then "FamilyBase" to control their data, its only $5, and you can monitor their phone droid and iphone, then install "Verizon Messaging" on all the devices that is free. i see what their texting without their knowledge.