Summary: Blind Deconvolution problem is a challenging task in several scientific imaging domains, such as Microscopy, Medicine and Astronomy. The Point Spread Function inducing the blur effect on the acquired image can be solely approximately known, or just a mathematical model may be available. Blind deconvolution aims to reconstruct the image when only the recorded data is available. In the last years, among the standard variational approaches, Deep Learning techniques have gained interest thanks to their impressive performances. The Deep Image Prior framework has been employed for solving this task, giving rise to the so-called neural blind deconvolution (NBD), where the unknown blur and image are estimated via two different neural networks. In this paper, we consider microscopy images, where the predominant noise is of Poisson type, hence signal-dependent: this leads to consider the generalized Kullback-Leibler as loss function and to couple it with regularization terms on both the blur operator and on the image. Furthermore, we propose to modify the standard NBD formulation problem, by including for the blur kernel an upper bound which depends on the optical instrument. A numerical solution is obtained by an alternating Proximal Gradient Descent-Ascent procedure, which results in the Double Deep Image Prior for Poisson noise algorithm. We evaluate the proposed strategy on both synthetic and real-world images, achieving promising results and proving that the correct choice of the loss and regularization functions strongly depends on the application at hand.