The authors define and investigate Effros metric \(\sigma_M\geq \rho\) defined on a bounded metric space \((X,\rho)\) by means of a semigroup \(M\) of surjections \(X\to X\) (a continuation and generalization of a paper by \textit{J. J. Charatonik} and \textit{T. Maćkowiak} [Trans. Am. Math. Soc. 298, 579-602 (1986; Zbl 0608.54012)]. The original Effros result on existence of ``small'' translations for homogeneous compact metric spaces [Ann. Math. (2) 81, 38-55 (1965; Zbl 0152.33203)] is then generalized for the above situation: \((X,\rho)\) has the Effros property with regard to \(M\) iff \(\sigma_M\) and \(\rho\) are equivalent. The results are applied to locally connected continua and local AR; for instance, local AR has the Effros property with regard to all surjections, no dendrite has the Effros property with regard to monotone maps.