Joi Ito, the MIT Media Lab director, has an interesting proposal for managing his "partial attention problem during meetings." Joi spends between 2-3 hours on email in the morning, and another 2-3 hours at night. In addition to that, he "must diligently triage email during the day." He also has a lot of meetings, and some of those meetings do not require his full attention. He needs to attend only to answer occasional questions or make decisions. So he proposes two kinds of meetings: "full attention" and "partial attention," which can be scheduled as such.
When someone signed up for a meeting, we would ask if they needed full attention and if so, they would end up in the "full attention slot" queue or get booked a month or so out when my next "full attention slot" was available. On the other hand, if all they wanted was for me to be available to provide opinions or make decisions as part of a broader meeting or if the person didn't mind my partial attention during meetings, we could book the meeting in a "partial attention" slot which could be scheduled sooner. I would use un-booked partial attention slots to catch up on email if no one wanted such a slot.
This feels a bit too clever by half and maybe difficult to communicate to a person not familiar with my problem.
The other idea that I had was just to ask at the beginning of a meeting, "do you want this to be a laptops closed meeting or do you mind if I keep my eye on urgent email and triage?" I'm not sure if everyone would ask for my full attention or if I'd have a selection bias where only people confident enough would ask for my full attention and that those people who really needed my attention but were too polite would end up with my partial attention.