Millimeter wave (mmWave) massive MIMO is now a reality but there is still room to boost the capacity of this technology by utilizing new techniques. One of the promising techniques that can be utilized together with mmWave massive MIMO technology is full-duplex transmissions. Compared to half-duplex systems, full-duplex transmissions can offer possibly higher data rate and lower latency, making it an exciting opportunity to further improve the capability of mmWave massive MIMO technology. Enabling full-duplex transmissions is hindered by self-interference (SI) that occurs at receive antennas of full-duplex systems. To overcome this interference, SI channel estimation is a crucial step to make full-duplex systems feasible. There are already several challenges such as high pilot overhead, pilot contamination, etc in channel estimation of mmWave massive MIMO systems, and due to the urge of SI channel estimation, utilizing full-duplex transmissions will further complicate channel estimation bottleneck in full-duplex mmWave massive MIMO systems. In this paper, we tackle the problem of channel estimation in full-duplex mmWave massive MIMO systems using deep neural networks (DNNs). We propose to share pilot resources between users and transmit antennas at the base station (BS) to reduce the pilot overhead associated with full-duplex transmissions. In order to overcome the interference caused by simultaneous pilot transmission by users and transmit antenna arrays, we denoise least square (LS) estimated SI and UE-BS channels with deep convolutional neural networks (CNNs). Thanks to the superior capability of CNNs, the pilot overhead of full-duplex transmissions can be reduced to the same pilot overhead of half-duplex transmission with a small performance loss.