Well.. kind of.
Mining is really thousands if not tens or hundreds of thousands of people competing to try to solve the same block. The one machine to solve it first gets the reward. (or if in a pool, it's given to the pool.) Over and over again- in the case of dogecoin, once every minute or so. The GP's statement about it being a massive waste of resources is unfortunately true - that's a lot of electricity for any popular coin.
> But over time, a secure network becomes valuable for itself, and the reward isn't necessary anymore.
Has this actually happened in any case? Logically if there is no value to be gained from maintaining the secure network (eg, 'mining' new transactions into blocks for confirmation), then many fewer people will do so. The fact that it's a secure network is great - but it doesn't seem enough to entice a critical mass of people to maintain it at a cost to themselves.