I have no experience with STI on a tandem. Our old bike is 3x8 Ultegra with bar-end shifters. 54t/44t/30t. The front shifter is friction (rear is indexed). For the most part, front shifts are pretty smooth. Rear shifts are immediate. I still find it helpful (sometimes necessary) to call out "shift". It may be possible to shift under load but still not a good idea. Shifting under load is hard on most everything (derailleurs, cogs & chain).
Our new Calfee is 2x11 Ultegra Di2. We adopted this because everyone that has it raves about it. I'm still getting used to it. My current view is that Di2 may be a solution to STI on a tandem - but not "better" than bar-end shifters.
Di2 would probably work well if one is willing to live with limited gear range. It doesn't work without compromises on our setup: 34f x 36r (low) to 52f x 11r (high). While it is possible to shift without informing my stoker, it doesn't work that great and still puts a lot of stress on the drivetrain.
Rudy & Kay have a lot of miles on their bar-end equipped tandem. They swear by it. Many here wouldn't think of going bar end but I can see how it may provide the best shifting for a wide gear range.
This is another case of: There is no "best" - only compromises that work best for each of us.