Can one use chess programs while on chess.com?  What if you're playing an unrelated game on a chess program while logged into this site?  You'd not be using the engine to help your online game, just playing an unrelated one against the computer.

I ask this because one day while on chess.com, I remembered I had a chess program I'd not yet installed, and then installed it.  Then I realized the program was running while I was online and got paranoid.  I wasn't even playing a game on the program, but both things were running at once.

I don't see the issue. Cheat-detection tends to be an offline process where the staff are sure-as-heck to crunch any suspicious game of yours and correlate it with an engine (hey, maybe several?) to prove/disprove any allegation of foul play. Basically => You don't have to get caught in the act, the minute you use engines for actual games, there's "hard" evidence => namely your game!!!

So I don't think you have to worry about a paperclip that might pop up (or for that matter a catholic nun with a wooden ruler) and chastize you for doing so.

That's good.  I like to multitask sometimes.

As long as you're being nice and not naughty ...

Of course.  My rating would be a lot better if I were cheating.  Plus, based on a chess.com analysis of two of my own games, it's a miracle I win at all - chess.com found all sorts of mistakes and blunders.