Experiments on the Ultimatum Game (UG) repeatedly show that people's behaviour is far from rational. In UG experiments, a subject proposes how to divide a pot and the other can accept or reject the proposal, in which case both lose everything. While rational people would offer and accept the minimum possible amount, in experiments low offers are often rejected and offers are typically larger than the minimum, and even fair. Several theoretical works have proposed that these results may arise evolutionarily when subjects act in both roles and there is a fixed interaction structure in the population specifying who plays with whom. We report the first experiments on structured UG with subjects playing simultaneously both roles. We observe that acceptance levels of responders approach rationality and proposers accommodate their offers to their environment. More precisely, subjects keep low acceptance levels all the time, but as proposers they follow a best-response-like approach to choose their offers. We thus find that status equality promotes rational sharing while the influence of structure leads to fairer offers compared to well-mixed populations. Our results are far from what is observed in single-role UG experiments and largely different from available predictions based on evolutionary game theory.