You can measure the speed of something towards/away from you by measuring the doppler shift of the signal (how much the frequency is increased or decreased compared to the expected frequency), and since the radio receivers will have to be very precisely pointed to get a good signal, you can also probably fine tune any estimates of position by wiggling the receivers around a little bit until you get the best signal. The signals are definitely getting degraded by noise etc, since it's so far away. That's why the communication speed is so slow, so they can make sure they got one bit before getting the next one. Some more mathsy details here https://space.stackexchange.com/questions/24338/how-to-calcu...
But by having a very big antenna, and knowing exactly what you're looking for and where, it can help to filter out all the noise and get out the proper data
It has a 3.7-meter (12 ft) diameter high-gain Cassegrain antenna to send and receive radio waves via the three Deep Space Network stations on the Earth. The spacecraft normally transmits data to Earth over Deep Space Network Channel 18, using a frequency of either 2.3 GHz or 8.4 GHz, while signals from Earth to Voyager are transmitted at 2.1 GHz.