This paper shows that the joint distribution of polarimetric intensity images is a multivariate gamma distribution in the case of coherent illumination with fully developed speckle. The parameters of this gamma distribution can be estimated according to the maximum likelihood (ML) principle. Different estimators depending on the number of available polarimetric images are studied. These estimators provide different ways of estimating the degree of polarization (DoP) associated to each pixel of the image. A performance comparison with estimators based on methods of moments shows the interest of the ML method for estimating the DoP of polarimetric images.