This reminds me of when my wife and first started dating long distance - Skype screen sharing movies was our jam
Of course, the biggest issue was that netflix offerings in the US were very different to those in the UK, so we'd spend about an hour where one of us tried to find dubiously legal ways to watch a programme the other could watch fine on netflix.
The skype is nice because it puts a thumbnail of their face that stays hovered over the netflix even while it's in fullscreen.