A network-assisted full-duplex (NAFD) system based on a cell-free (CF) massive multiple-input, multiple-output (MIMO) framework has been proposed to satisfy the demands of higher data transmission rates and efficient communication. However, pilot contamination may occur due to the reuse of pilot sequences in a massive MIMO. With this consideration, we raise an asynchronous channel estimation method based on an uplink and downlink time-shifting pilot-sending scheme, which is able to avoid pilot sequence reuse when obtaining channel state information (CSI), while the data signals could be transmitted simultaneously at the same frequency. The transmission processes of the proposed method above are divided into three phases, including pilot phase, estimation phase, and data phase, in chronological order. When the uplink is in pilot phase, the corresponding downlink is in data phase and vice versa. After the channel state information estimation, both uplinks and downlinks are in data phase. The maximum ratio combination (MRC) receiver and the maximum ratio transmission (MRT) precoding are adopted in the uplink and downlink. The closed-form expressions are derived based on large-scale random matrix theory. We compared our asymptotic results with practical results in simulation, and find that they are well matched. Moreover, the proposed method is superior to the normal time-division duplex (TDD) system.