Depends on the argument, and the person. I had a huge blow out with a coworker friend of mine, where I basically told her to get her shit together because being friends with her wasn't great because she just cared about herself, essentially (we'd been friends for a long time) ... it's an unspoken thing now that we aren't "best buds" but we're still cordial and friendly (the conversation was never brought up again, but I can tell she made some effort to change, for her sake I think that's great.) The weeks immediately after though were sort of awkward, but we're both pretty professional.

Personally, my style is to talk things out. However...

If it's a coworker you're not hardly around a lot that often except work, you're not really acquainted, and the argument or subject matter is tense enough: don't talk to them - go straight to your supervisor and say something - so the supervisor can talk to them, and then maybe resolve the situation with a group chat if necessary. That way you can avoid the "he said she said" stuff.

That's just how I would do things. But sometimes I'm an idiot, take it with a grain of salt.