For that matter, the natives of the New World had diseases of their own which were brought back to the Old World by sailor on the explorers’ ships. (Syphilis, for example, is believed to be one such.)
It’s just that these diseases caused much fewer deaths back in the Old World, because the people there had a stronger immune system (due to more people living in more crowded (thus dirtier) conditions). Or possibly they did cause as many deaths, but it wasn’t as noticeable given the much larger populations in Europe – just a few more people amongst the many that died every day of various diseases & plagues.