An autoencoder has the potential ability to compress and decompress information. In this work, we consider the process of generating a stego-image from an original image and watermarks as compression, and the process of recovering the original image and watermarks from the stego-image as decompression. We propose embedder and extractor neural networks based on the autoencoder. The embedder network learns mapping from the DCT coefficients of the original image and a watermark to those of the stego-image. The extractor network learns mapping from the DCT coefficients of the stego-image to the watermark. Once the proposed neural network has been trained, the network can embed and extract the watermark into unlearned test images. We investigated the relation between the number of neurons and network performance by computer simulations and found that the trained neural network could provide high-quality stego-images and watermarks with few errors. We also evaluated the robustness against JPEG compression and found that, when suitable parameters were used, the watermarks were extracted with an average BER lower than 0.01 and image quality over 35 dB when the quality factor Q was over 50. We also investigated how to represent the watermarks in the stego-image by our neural network. There are two possibilities: distributed representation and sparse representation. From the results of investigation into the output of the stego layer (3rd layer), we found that the distributed representation emerged at an early learning step and then sparse representation came out at a later step.