In this paper, three infinite families of locally 2-arc transitive graphs are constructed, which are vertex-in transitive, regular and all vertex stabilizers are conjugate. To the best of our knowledge these are the first infinite families of graphs with these properties. In particular, they are semi-symmetric. It is then shown that the only locally 2-arc transitive graphs admitting a Ree simple group are (i) the graphs in these three families, (ii) (vertex-transitive) 2-arc transitive graphs admitting a Ree simple group, previously classified by the first and third authors, and (iii) standard double covers of the graphs in (ii). This is the first complete classification of locally 2-arc transitive graphs for an infinite family of simple groups. (C) 2004 Elsevier Inc. All rights reserved.