What device model are you referring to? If they are using iMessages, Hangouts, Facebook messenger, or any other application that uses data to transfer messages between devices, the message history won't be displayed in the text message log on the MyVerizon account online.
I'm most definitely NOT a VZW employee. If a post answered your question, please mark it as the answer.
Anything sent through the Verizon Wireless Network (Calls, messages etc) will display on the bill and unbilled usage online. The only thing any can see is the Time and phone numbers and locations. They cannot see the actual message. If you use a 3rd Party application as Ann154 stated those are not sent through the network and therefore are not recorded.