A possible form of hearing the presence of a Russian stealth submarine is focusing on the electronic systems on-board, rather than trying to scan for noises commonly made by the diesel engines, propellers, or sonar detection. Since the diesel generators would likely not be running while the submarine is on its mission, normal noises would not be heard. There is still the possibility of hearing the propeller while the submarine is moving. Additionally, scans for water displacement could determine the presence of the submarine. This could be computed based on water currents that change when the submarine passes by buoys or drones.
Alternatively, this submarine scan can be replaced by scanning for radio frequencies and common noises made by computers, while in reality the sensing equipment would need to be fine-tuned since the noises are so quiet when compared to the noise of the water outside the hull.
There is also the possibility of detecting the presence of the submarine using lasers that can penetrate through water and reflect back to tell the submarine hunter what the shapes of the objects are in the surroundings. Also, underwater cameras with powerful lights that can penetrate through a long distance under water could pick up the optical signature of the submarine, especially since it would be operating so close to the surface.
If multiple submersible drones are used, and at least 2 to 3 of them ping with sonar detection at the same time, the submarine would not likely be able to mask its signature using the stealth technology. This is likely the most probable solution to searching for a Russian stealth submarine.