The actual messages must be viewed on the phone and the numbers that texts were sent from will be on your bill in My Verizon in detail. What device is your child using?
You CAN see it. Simply look at the phone. If there are gaps on the phone from where your online MyVerizon acct says there should be texts, simply don't return the phone until your child can respect the rules for the phone.
if your child has an iPhone and is messaging other iPhone users then you won't see their conversations unless you physically look at their phone.
Of course, you are assuming iMessage has been activated on both ends of the conversation. If not, messages to/from other iPhone users will be sent/received as SMS/MMS messages just as to/from any other platform and NOT as iMessages.
Not all iPhone users use iMessage as it will use data vs free unlimited texting available in the More Everything plan. Another reason I don't personally use iMessage is because I like to access my messages on multiple devices from within Verizon Messages + which you just can't do when you have some messages sent as iMessages and others sent as SMS/MMS messages.