In almost any geostatistical analysis, one of the underlying, often implicit, modelling assumptions is that the spatial locations, where measurements are taken, are recorded without error. In this study we develop geostatistical inference when this assumption is not valid. This is often the case when, for example, individual address information is randomly altered to provide privacy protection or imprecisions are induced by geocoding processes and measurement devices. Our objective is to develop a method of inference based on the composite likelihood that overcomes the inherent computational limits of the full likelihood method as set out in Fanshawe and Diggle (2011). Through a simulation study, we then compare the performance of our proposed approach with an N-weighted least squares estimation procedure, based on a corrected version of the empirical variogram. Our results indicate that the composite-likelihood approach outperforms the latter, leading to smaller root-mean-square-errors in the parameter estimates. Finally, we illustrate an application of our method to analyse data on malnutrition from a Demographic and Health Survey conducted in Senegal in 2011, where locations were randomly perturbed to protect the privacy of respondents. We conclude that our approach based on the composite likelihood is a feasible and computationally more efficient alternative option to existing likelihood-based methods that deal with positional error in a geostatistical context.