Burden of Proof

While the burden of proof, or who has to back up their theories is the group or person with the idea which is not mainstream, isn't there some proof needed from the people on the other side as well. For example, when the earths population believed that the world was flat, the burden of proof fell with the believers of a round world. But didn't these flat-world believers ever have to prove their theory? Or were they just assumed right with the information they had at the time and because nobody had a better theory? The same can be said for people who believe aliens exist or there is some sort of life on distant, or maybe not so distant planets. While the burden of proof lies with the UFO-folk, doesn't there have to be some sort of concrete evidence that there is no life as well? You can't just say because we can't find it it's not there. Right?