It depends on the watch. If the watch is able to connect to Wi-Fi by itself, such as a Samsung Gear S2 Classic, then it will pull in notifications, as long as the phone is connected to "a" network, not necessarily the same network.
Having a wearable that would notify you when you leave your phone behind would reduce this tremendously while increasing "smartwatch" sales. We should be able to set a notification for a bluetooth disconnect and another notification for a total disconnect.
Android Wear had had this ability for years as per the Feel The Wear app. This is the main reason I'm hesitant to keep the Gear S3 after the 14 day return policy ... even though it is the best watch ... based on a platform that is doomed to fail (think beta-max) ... TIZEN.
So then, can this be done on the Gear S3 or is this a "dumbwatch"?