No, the engines are not downloaded to your PC, but to your browser. The first time you pick the engine it will download it and store it in your browser so you dont have to download it everytime you come back onto chess.com. The moment you close the tab for chess.com, the engine stops running.
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in the game analysis settings, when I click on the dropdown menu for the engines, it states (download) next to the various options. I may be a dinosaur, but does that mean each time I run the site's engines, they're downloaded to my PC? It's just that my PC seems to get stuck running with a very high fan speed even after I close the site down every time I analyze a game.