If I turn cellular data off while I'm using WIFI, I don't receive any messages including voicemail messages. I have enabled iMessage, SMS, and MMS messaging but still do not receive any messages until I turn cellular data on.
While connected to Wi-Fi, you should definitely receive iMessages, especially if the sender is sending the message to your Apple ID, however, your Apple ID must be listed in the "Send & Receive" option in the Messages settings. As for SMS, MMS and voice messages, I'm not sure. It may be a limitation. Check with your wireless carrier for confirmation.