I always watch them Fullscreen and simply switch between the video and Blender (might depend on your operating system how easy that is...).
Also, first watch the whole video without using Blender, and then watch it again in parts and follow along, switching between tutorial and Blender.
Yeah I do the same thing, I watch it in full screen and ALT + Tab to switch to Blender and again to switch to the video. That is when I don't have a second monitor plugged in, which most people do, but not everyone can. And again as Martin said, most of the time I watch all the video first, so I watch the whole course as if watching a movie and I always get the feeling that I got it all. But then when I'm finished I try to do the stuffs I saw only to find out I have gaps on my knowledge, in other words I forgot most of what I saw, but the sense of it all is there, so I got back and watch it again only this time I'm following along with Blender open, and this second time around the lessons solidify in my head strongly.