We live completely parallel lives, to the point where our kids rarely talk to their dad when they're with me, and vice versa. That means I usually go a week at a time without talking to my babies, which is harder some weeks than others, but the flip side is that I don't have to interact with him on my weeks with them.
How does it work? We just trust each other to keep the kids healthy, safe, and get them to school on time. That's it. I think he's overly strict with them, and he got peeved when I let our 11-year-old read Twilight and The Hunger Games. He doesn't allow sugar cereal; I toss Pop-Tarts at them in the car on the way to school. We go to churches of different denominations. Santa doesn't visit his house, and the kids aren't always in bed by nine at mine.