It is completely and utterly inaccurate to claim that the visits will not be logged because you're using two different domains. This is the wrong answer.
As I already stated above quite explicitly, the visits will be anonymous on the other domain because those web sessions will not be associated originally. If the web sessions become associated (if you send someone a link at the temp domain, for example), those will be known sessions. Either way they are being logged and it's incorrect to usggest otherwise. Support should know better.
Jamie Gregory please withdraw this answer, as it's going to be very misleading to anyone who searches in the future.
Munchkin can be used on as many domains as you want. Munchkin associations are not shared out-of-the-box but that isn't the same question.
We are about to embark on taking our site to a new wordpress site. I'm being proactive and making sure I supply our new website company with all relevant information. I noticed in Jamie's screen shot above her API Configuration is disabled, but her private key is set. Mine is not set and disabled. See screen shot. Is this an issue, should ours be set? If so, how do I do that?
It's better to open a new thread so later searches are easier (remember, there can be only one official answer per thread).
You don't need to supply the Munchkin API key at all to your provider unless they're going to use that API. In the other screenshot it's been enabled, then disabled again.