In my experience, no! I actually wondered this myself. I started one in Georgia (did not turn out well :') ). Tasted very different from the one I bought here, which I also had in Georgia. I went west to California. My roomie went north to Mass. Hers got moldy and trashed eventually, but still tasted the same. Mine still tastes the same way it did in GA, a few thousand miles away. It's been a few years now. Flavor hasn't changed much.
From a science standpoint, the symbiotic culture of yeast and bacteria of a well established starter should be robust enough that environmental microbes are outcompeted and can't get established. This is also why a healthy starter will never mold even though it's a heap of wet nutrients sitting on your counter (imagine if you left out a soggy loaf of bread, which is pretty much a starter without the culture, and how fast it would mold).
Doen't the starter conform to the local environment after a feww weeks of using and feeding?