When sending iMessages to someone they received them only when when connected to internet

Hi, I have a problem with imessages. I noticed that when I’m sending imessage to someone, it is delivered, but that person doesn’t have internet and reseiving my messages, only when he connects. Also I can’t send as text message. Need to go to settings and set imessage as turned off, then I can’t send anything at all. Have to delete conversation and start a new one. Does anyone have this problem? How to solve it?

[Re-Titled by Moderator]