Do you send chat messages when not connected to the internet, either via a WiFi or a mobile data connection? Chat in Samsung Messages uses a similar system to iMessage on iOS, in which a message will default to SMS or MMS when not connected to the internet, again via WiFi or mobile data.
Also, Chat, as with iMessage, only works with devices in the same ecosystem (Samsung for Chat, Apple for iOS), so are the devices you're sending the messages to Samsung devices? Cause unfortunately Chat doesn't work across all Android devices, and DEFINITELY NOT WITH APPLE DEVICES.