Pillow on the Apple Watch should require the iPhone only if you try to start a sleep session with at least of the following options checked:

These options use the iPhone, so connectivity is required in order to start a sleep session. During the sleep session you can turn off your iPhone, but you have to turn it on if you need to access your sleep sessions in the morning. This applies only to sleep sessions that don't have any of the options mentioned above.

If you don't use an Apple Watch then your iPhone must be turned on in order for Pillow to perform sleep tracking.