Counting tasks with overlapping and occluded targets are often tackled by means of neural networks outputting density maps. While this approach has been proven to be highly effective for crowd-counting tasks, it has not been exploited extensively in other fields (like fruit counting). Furthermore, this approach has never been used to infer the shape or the size of the recognized objects. In this paper, we present a novel deep learning-based methodology to automatically estimate the number of grape berries present in an image and evaluate their average radius as a double output of the network. For the model training, we employ a public dataset consisting of 300 vines images, where each berry center has been dot-annotated. Since the dataset does not directly provide information about the berry radii, we first develop a numerical optimization methodology to calculate the radius of the berries, by exploiting the dot annotations, some prior knowledge (berry maximum size), and a current state-of-the-art segmentation model. Then, we employ the combined information (berry center and radius) to train a custom neural network that outputs two density maps, from which we infer the number of berries in the image and their average size.