I've used home covid tests on two occasions, and both times the results were negative. Sometime later, I found out that I had used the tests incorrectly. You are supposed to wait five days after possible exposure before you test!
It's possible that I was actually covid positive but asymptomatic. That's important, because even if you're asymptomatic, you can still spread the virus to other people!
I do a test if I have any symptoms that could possibly be covid. I also did one before attending a party in May. I have not been knowingly exposed to anyone who has covid so I haven't taken one in the situation you describe.
I have a co-worker who was in that situation and he took tests daily after being informed he'd been exposed. He didn't test positive until day 5. On the other hand, I have other coworkers who tested positive a day after arriving home from a company meeting and learning that people they met with had tested positive the day before. All may have been exposed a few days' prior, though.