First of all, I have made my research and I cannot figure this out on my own.
I set up the emergency contact, including email address and phone number with the "+1" prefix. At this stage everything seems to be ok. Testing the feature by triggering an alert from the menu seems to only send an email to the recipient, no text message whatsoever - not even the "I'm OK" text message. I tried with different contacts. Please note that I haven't tried during an activity.
I am not sure if the issue comes from the watch or the settings on my phone (iPhone 8Plus with the most recent iOS version).
If someone could help me sorting this out, It would be great 
Thanks a lot.
