Not at all! Angel Watch ™ is its own phone, so just like any mobile, you have your plan, they have theirs.
For customers in the US and UK, Angel Watch™ can have its own fully supported SIM card.
The plan is month-to-month (no contract) so you can cancel anytime and there's no chance of overage charges.
So with the watch having its own plan with its own number, sufficient minutes (1000), and data (500MB) or Unlimited.
There is no need for you to change your network personally and calls would just come out of your plan's included minutes.