To my knowledge, Voyager accessed the alien network only twice before it collapse, to send the Doctor and receive Starfleet's response. However, it can be hacked: In the STO timeline, Sela used in 2403 to contact Hirogen in the Beta Quadrant to ally with them. Btw, by 2409, every single ship and shuttle class can be equipped with QSD, including Federation, Klingon, Terran, Cardassian, Ferengi, Dominion and Breen starships.
I thought dialog indicated that the crew had been receiving letters for a couple of weeks over the network? Or am I thinking of the hyper-subspace relay that was introduced in Pathfinder
? I know that once the events of Pathfinder
occurred, Voyager was in contact with the Alpha Quadrant on a monthly basis!