Both users must be on the same carrier "usually" unless you are using a special SMS app like "WhatsApp" that can provide that type of information.
ie, on Verizon, you always get a 'Sent' time stamp, and then a "Received time stamp" regardless of what SMS app you use.
I have no direct experience with other carriers, but anytime I send SMS to a user on a different carrier, all I will ever know is "time sent".....
I only use WhatsApp to talk to my son, he is a security freak.