AFAIK MMS & SMS go over the cell towers (hence the terrible compression if you send a video, for example). If you use chat such as iMessage (Apple) or RCS (Google), those use the internet and would be downloaded after you connected to an internet source.
If you use RCS and it can't send, it will usually ask if you want to use SMS/MMS instead, which happens frequently at my office b/c my signal on Verizon is terrible in our building.