It is possible to use conventional cranks in a crossover setup, but you will need 3 right pedals and one left. With some pedals it is possible to interchange the axles, so that the pedal which is shaped for the left foot can have the right-threaded axle, and vice versa. This is not generally possible with clipless pedals. Most store-bought tandems are set up this way. Front drive is the less common arrangement, using a loooong primary chain that runs all the way from the captain's crankset to the rear hub.
Front drive has much to recommend it, particularly on a home-built.
It puts less torsional stress on the crossover bottom bracket. Front drive is a bit heavier, due to the extra length of chain required, and you also need to take care that the stoker crank does not strike the chain. The stoker's right crank has two chains running to it, the synch chain going forward to the pilot's crankset, and the primary chain running back to the rear hub. Stress on the stoker's bottom bracket is actually less than on a solo, and much less than on a crossover setup. The only downside of single-side drive is that it complicates the use of a front derailer.