Welcome to the community!
What phones are you using? Are the messages sent to you text messages (ie SMS or MMS) or is it possible they are iMessages or rich communication services (RCS) messages (see also here)? Both iMessages and RCS messages would require data to be sent and received. Since your plan does not include data, that could account for why you only get them when connected to Wifi. Is it possible that is a reason why you're only receiving them when on Wifi?
**edit** In order to send and/or receive MMS, you would need to enable data on your device. While you do not need to have a plan with data in order to send/receive MMS, data does need to be enabled. However, both iMessages and RCS messaging would require a plan with data.**
Hope this helps 😀