Signals used for communication use a few different systems, which would mitigate overall frequency change due to red and blue shift.

In analog systems, FM is the most common. This uses a carrier wave, and alters its frequency to impart information. That entire carrier wave would be higher or lower in frequency, but once you pick it up, the changes to it representing the information would still be fine. Imagine having to tune your radio to 94 MHz instead of 97MHz, but still receiving the same signal.

In digital transmission, it’s very common to have a “clock” pulse embedded in the signal. This clock effectively tells the receiving system what frequency it’s at, and where the data packet begins and ends.

Neither such system would have an issue with a red or blue shift.