They did: syphilis! But the Europeans had far more diseases to share because there was far more animal domestication going on in the Old World. And most of our diseases came as a result of that animal domestication, so they had already spread through the population which developed immunity in the millennia between the first human infection and the Columbian Exchange.
> Usually the smallpox theory is presented in a way that removes agency and culpability from the conquerers. It's always struck me at remarkably convenient and quite unbelievable; they were an idyllic people in some welder land without their own diseases, oh really now ... we're talking the Caribbeans here
Typically I hear "the smallpox theory" presented as "Europeans killed 90% of Native Americans including by disease" as though Europeans collectively set out to exterminate Native Americans. To be certain, there was a lot of brutality and genocide and even some deliberate spread of disease, but no European could have credibly believed that the disease would spread throughout the new world to such effect.