Let (X, Y) = (RU (1), RU (2)) be a given bivariate scale mixture random vector, with R > 0 independent of the bivariate random vector (U (1), U (2)). In this paper we derive exact asymptotic expansions of the joint survivor probability of (X, Y) assuming that R has distribution function in the Gumbel max-domain of attraction, and (U (1), U (2)) has a specific local asymptotic behaviour around some absorbing point. We apply our results to investigate the asymptotic behaviour of joint conditional excess distribution and the asymptotic independence for two models of bivariate scale mixture distributions.