One of my professors years ago came from a photojournalism background and really drilled the 'both eyes open' ethos into our heads... and when I say drilled, I mean he would have us doing literal drills in order to get our minds around the idea and eliminate the 'bad habit' of closing one eye as soon as we put the viewfinder to our eye.

What he had us do started out pretty simple... Over the course of a quarter the progression went something like this:

1. Initially we just sat in the studio/classroom and practiced putting the camera to our eyes but NOT closing the opposing eye... I don't mean that we did that a few times... I think we spent 3 or 4 entire class sessions (the equivalent of nearly 3 or 4 hours) on the task. This was all about muscle memory, and in many ways it seemed very similar to the sorts of drills you'd see soldiers doing in boot camp as they learned to raise and aim their weapons with control and speed.
2. The next thing he had us do was close our viewfinder eye and do all our aiming with the eye that wasn't looking through the viewfinder. We must have spent a week on that alone, and it felt really strange at first, but it really trained us to not simply ignore the 'non-viewfinder' eye as soon as the camera is up in our face.
3. Then we practiced aiming our shots with the non-viewfinder eye with both eyes open. It's really interesting to feel yourself mentally shift focus from one eye to the other, and after a bit it became an almost instant and unconscious action... Ultimately, that was really his point in having us do this exercise (and all of them, really).
4. After that we spent weeks out on the quad taking pictures of people. He'd let us kinda be doing our own thing and then randomly he'd shout out to 'take a picture of the guy in the green hoodie' (or whatever the student he happened to be looking at was wearing). The instinct for most of us at first was to pull the camera away from our faces and look around for the person he was talking about... But over time we got good at swiveling around with our eyes up to the viewfinder and both eyes open searching for the photograph but never removing the camera from our eye... I'm sure it was quite a sight to see 25 photographers all simultaneously turn wherever they were and take a picture of a person. Later I found out that he actually recruited people to walk the quad during our classes, which I suppose makes sense in terms of not scaring random people walking to their classes. :-)
5. Finally, the thing he had us work on was putting it all together, that is getting to a point where we were composing shots well on the fly, never removing the camera from our eye, essentially using our 'free' eye to find the next shot while we were still working on the current one with our viewfinder eye.

So that was the progression of drills he took us through, and it's a progression of drills that I still use with my own photography students (and I still drill myself... Mostly when I add a piece of new gear, a lens, or a new camera body). In terms of the advantages, certainly there's less fatigue in not having to shut one eye, but I think the real advantage is in having situational awareness and the muscle memory to compose photographs happening around you very quickly and in a precise way. There are plenty of wedding photographs that I take 'on the fly' that I almost don't remember taking because I'm moving around the setting with both eyes so quickly, composing as I go, and capturing the moments...

As an aside, all of this is made easier with the control that back-button focus gives the photographer, and I 'never leave home without it.' :-)