Evan Solomon, an engineer at Medium, describes his efforts to "ignore all distractions from my phone when I’m with other people:"

There’s an expanding set of distractions in most people’s lives. We’re also getting better at building distracting tools that make it remarkably easy to ramp up your own morphine drip of incoming messages, likes, faves, and retweets. There is also a lot of cultural pressure to be distracted. The New York Times published a great op-ed called The ‘Busy’ Trap about a year ago….

It took about a week of practice before I could actually care more about listening to the people I was with than thinking about whatever else I might have been doing. I didn’t think of it as practice, but in hindsight that’s exactly what it was. I was practicing breaking a habit until one day I had a new habit….

I haven’t missed anything that really mattered. Not one single thing. No one has complained; I doubt anyone has even noticed, and that’s the idea. Your mileage may vary, but unless you’re an on-call doctor and decide to silence your pager at dinner I bet your results would look a lot like mine.

This meshes very well with what digital Sabbath practitioners told me: even on a month-long e-mail break they didn't anything, and certainly could go a day without fear of missing out.