Welcome to the community!
What phone 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 you appear to only be able to receive messages using data, that could account for why you only get them when connected to Wifi. SMS and MMS do not get sent via Wifi (unless the device is set-up for Wifi-calling). Is it possible that is a reason why you're only receiving them when on Wifi?
Hope this helps 😀