After having left the heliosphere, Voyager 1 and Voyager 2 continue to travel through interstellar space. The Pioneer 10, Pioneer 11, and New Horizons spacecraft are also on paths to pass the heliopause. These spacecraft have communicated with the Deep Station Network (DSN) radio antennas in order to download scientific data and telemetry data. Outward transmissions from DSN travel to the spacecraft and beyond into interstellar space. These transmissions have encountered and will encounter other stars, introducing the possibility that intelligent life in other solar systems will encounter our terrestrial transmissions. We use the beamwidth of the transmissions between DSN and interstellar spacecraft to perform a search around the past and future positions of each spacecraft obtained from the JPL Horizons System. By performing this search over the Gaia Catalogue of Nearby Stars, a catalog of precisely mapped stars within 100 pc, we determine which stars the transmissions of these spacecraft will encounter. We highlight stars that are in the background of DSN transmissions and calculate the dates of these encounters to determine the time and place for potential intelligent extraterrestrial life to encounter terrestrial transmissions.