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Local energy markets (LEMs) aim at building up local balances of generation and demand close to real time. A bottom-up energy system made up of several LEMs could reduce energy transmission, renewable curtailment and redispatch measures in the long-term, if managed properly. However, relying on limited local resources, LEMs require flexibility to achieve a high level of self-sufficiency. We introduce demand response (DR) into LEMs as a means of flexibility in residential demand that can be used to increase local self-sufficiency, decrease residual demand power peaks, facilitate local energy balances and reduce the cost of energy supply. We present a simulation study on a 100 household LEM and show how local sufficiency can be increased up to 16% with local trading and DR. We study three German regulatory scenarios and derive that the electricity price and the annual residual peak demand can be reduced by up to 10ce/kWh and 40%.