アメリカのキャリアがどうなのか調べてみた。
AT&Tもsmsとmmsは駄目みたい。

To send and receive SMS/MMS messages on your Apple Watch Series 3, the synced iPhone must be powered on and connected to the AT&T network.

http://about.att.com/story/iphone_8_apple_watch_series_3.html