Actually, that depends on your phone. For example, my iPhone has to have data on in order to send and receive MMS messages. The only way around that is when I enable Airplane Mode on my iPhone, connect to Wi-Fi while Airplane Mode is enabled, and use Wi-Fi Calling to send and receive MMS messages. I know because I have tried to send and receive MMS messages with data turned off, but it failed.
You pose a good question, and I can definitely clear this up for you. Picture (MMS) Messages do require data in order to go through. Text (SMS) Messages do not require data, and can be sent and received with data turned off.