I just tested the network relay, and I think it actually always works through the internet. The message finds a user who has internet, and their app tells goTenna’s server to send an SMS to the target. I tested it on 2 phones, and got a text from an unknown number, on my behalf. It makes sense to make it this way, cause it would be invasive to use another person’s text messaging to do it. Although, it’s using their internet connection, which is not as bad, but IMO the “relayer” should have the option to opt out of it.
Also, even though technically Android is able to let third party apps send SMS, iOS isn’t, so this way both can work.
It’s not clear from your phrasing, but just in case, the gotenna doing the final relay has to be connected to a device with internet, can’t be by itself.