Bar-end index shifter (aka, Barcon) on Captain's right hand drop-bar end.
Rarely used, so stoker-captain coordination of brake operation is not something we have much time to practice. Therefore, it's just less complicated for me to manage the application of the drag brake on those rare occasions.
For some teams, giving the stoker control of the brake does "involve them" more in the operation of the tandem but Debbie's never indicated she feels she needs a greater role in operation of the tandem.
Regarding bar-end, vs thumbshifter vs extra brake lever... what ever works. If you use bar-end shifters instead of STI/Ergo or downtube shifters, the bar-end shifter isn't viable for the captain. This is why the thumbshifters were originally adopted for use with Arai drag brakes on tandems. The downside of thumbshifters is that they clutter up the tops of handlebars. As far as stokers using a mountain bike or drop-bar brake lever, again, that's a personal choice thing. The drop-bar brake lever on a stoker bar could certainly be problematic since it's likely that it would interfere with the captain's leg. The mountain lever would certainly work. To be honest, I don't think I've personally seen a Arai drum brake operated by a stoker using a mountain bike brake lever. I've seen lots of bar-end shifter operated Arai drum brakes and a few thumbshifter operated set-ups.
A lot of the different tandem set-ups for brakes, shifters, and handlebars take on a regional flavors dependent on what set-ups are favored by the folks who sell tandems If the regional flavor suits your needs go for it. Ultimately, you'll need to figure out what works best for you.