In regression analysis, when the covariates are not exactly observed, measurement error models extend the usual regression models toward a more realistic representation of the covariates. It is common in the literature to directly propose prior distributions for the parameters in normal measurement error models. Posterior inference requires Markov chain Monte Carlo (MCMC) computations. However, the regression model can be seen as a reparameterization of the bivariate normal distribution. In this paper, general results for objective Bayesian inference under the bivariate normal distribution were adapted to the regression framework. So, posterior inferences for the structural parameters of a measurement error model under a great variety of priors were obtained in a simple way. The methodology is illustrated by using five common prior distributions showing good performance for all prior distributions considered. MCMC methods are not necessary at all. Model assessment is also discussed. Results from a simulation study and applications to real data sets are reported.