You can have cross contamination. Once I had my sourdough starter going right next to a batch of cheddar and the cheddar didn't work. I learned my lesson on that one. But most of the time, I don't have issues with it.
What I now do when making cheese is add the culture when the milk goes in the pot, even before it is up to temperature. This protects it from cross-contamination (not 100% but I haven't had a problem since I started doing this). Here's a video I made on youtube about it.