They know the answer already, and are probably both trying it.
In US terminology, since that's the language I know, they try for "competition" rather than "conflict". The difference being whether they respect each other's sovereignty for the most part while trying to bury the other, and don't take straight-up military actions.
To achieve this, you provide a long series of "offramps" - opportunities to pause and de-escalate - on the path between peace and MAD, and ensure there is no benefit to either party to do any specific escalation. Mistakes will happen, both deliberate and accidental, but they're very unlikely to all happen at the same time, so even if things get tense there's offramps left, and game-theoretically they will take one because nobody wants a full-scale nuclear conflict.
Hmm. What areas? AFAIK they're on all the same continents as humans, and a few species get as far north as places like Canada. I guess Ireland famously has no snakes, so maybe there. I'd expect northern Canada and Ireland could both get dragon myths by import from distant lands.
So, that's my attempt at an explanation.