Anecdotal of course, but in the case of a common cold (as opposed to one where symptoms are unusually severe, or lasts too long) no-one in my family takes medicine (other than tea with honey and lemon), and the cold goes away just fine. I get the strong impression this is true for most of Europe.

A little less anecdotally, a cold is a viral infection, so medicine will mostly target only its symptoms, not the underlying cause. If the symptoms are bearable, then this looks like a bad idea to me - I'm inclined to think the symptoms are there for a reason.