That's some great questions. I think there are several important considerations.
One is that what we see is not the total transfer of deseases but rather the result of one or two major pathogens. It's quite possible, likely even, that there was an equal exchange of pathogens, a two way street. However, on one side there were one or two really horrific examples, smallpox as an example, that spread easily, rapidly and with devestating results.
Second, initially there were only very small colonies of Europeans. Many of them did die out and we simply don't know the reasons.
Finally, while the possiblility of transmission was wide open in the New World, the transmission in the other direction, from the New to the Old, was far more restrictive. There were few peoples that made the journey both ways, from Old to New and back again to the Old, so the evidences of any such infection would be most likley here.
Aslan is not a Tame Lion
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