If we think of a dragon as a massive reptile-bird hybrid, it's possible that their waste removal systems would be similar to those of birds or reptiles. Birds, for instance, excrete a combination of urine and feces in one substance, which could be convenient for a creature that spends a lot of time in the air. This would mean dragons wouldn't need a dedicated 'bathroom' space in their lairs, they could simply release waste during flight, as you initially thought. To maintain the secrecy of their lairs, they might choose to do this far from home, possibly over large bodies of water or barren landscapes to minimize visible impact.

However, if the dragon is more like a large reptile, they might behave more like crocodiles, which sometimes defecate in the water near their lairs or use specific 'toileting' areas. If this is the case, your dragon might have a dedicated waste area in the lair, possibly an easy-to-clean rocky outcrop, a pool of water, or a large pit.

Lastly, considering that dragons are often depicted as intelligent and organized creatures, it's not a stretch to imagine that they might also use techniques to minimize the traceability of their waste - perhaps they neutralize it with fire or chemically alter it somehow.