Why do we need testesterone and oestrogen?

Bodily androgens produce oestrogen in females and testesterone in males, being the primary gender hormones. Yet I have heard that females have testesterone and males have oestrogen in significant quantities. Why do we need testesterone and oestrogen? And why do males and females have the others' hormones? And what would happen if we had neither testesterone nor oestrogen?