Things do not become non-kosher simply through exposure to non-kosher items. There are two basic rules to know:
A non-kosher item can only render something non-kosher if it transfers some of its [significant; not subtle] flavor to the thing. This is a very basic concept, and is found in Shulchan Aruch YD 98:1. (How we ascertain if flavor has been transferred is a matter of disagreement. The Shulchan Aruch, following the ruling of Gemara, rules that we ask a non-Jew to taste the thing. Rema says the custom is not to rely on that and thus to require 60:1 whenever the matter is uncertain.)
Even if flavor has been transferred, if the mixture as a whole is any less pleasant tasting because of this added flavor, it does not render the thing non-kosher. This is also a basic concept, and is found in Shulchan Aruch YD 103:1.
These two points are the Halachic avenues which provide the basis for people who will drink coffee at any coffee house. Without getting involved in specific, technical arguments (a number of which can be found here under the question "Doesn’t the Chicago Rabbinical Council advise against many of the beverages which you recommend?") the basic idea is that you never, ever taste any meat/fat flavor in your coffee, nor would most people want to, so in essence you have both rules 1 and 2 working for you.
I am not denying the fact that the other side may have some technical arguments which are cause for their disagreement with this conclusion, though they certainly do not argue fundamentally on the above two points. This is just to give you a basic idea why someone's rabbi might allow them to drink in any coffee shop even though there are non-kosher items processed at the same facility.