"Dry" and "sunny" are two different parameters; you can control the moisture retention with potting techniques and composition of the planting medium. I can't think of too many Begonias that don't prefer to dry out between waterings; I heard someone on a gardening podcast yesterday refer to begonias as succulents. It's never occurred to me, but they do have a lot of water-storing adaptations. As far as light goes, most will do well with bright light, but you'll need to experiment to keep the foliage from burning. The semperflorens types grow here in full, blazing sun, especially the varieties with bronze leaves. I wouldn't try the rex or other rhizomatous varieties with big leaves in too sunny a location, and some tuberous varieties might object more to the heat generated by such a location than to the light itself. You'll also need to make sure that humidity remains high, or spider mites can become a real problem indoors.

My tuberous begonia was outside all summer on my balconey last year. Got full sun from morning until around 1-2 pm. No leaf burn but I did notice that the flowers always bloomed down amongst the foliage. After a short rest, it started blooming indoors late winter. The flower stalks reached beyond the foliage so apparently high light can impact flower stalk length.