That is actually quite easy.
Step 1 :
Make sure to launch the Chrome instance hosting the page you want to inspect (in your case the primary host) with the following additional parameter :
--remote-debugging-port=9222
Step 2 :
If the Chrome instance where you would like to have your Developer Tools front-end was on the same machine, you would then just have to visit the following url to get a list of the inspectable page :
http://localhost:9222
Though, as you want to run the front-end on a different computer, you should target your primary hostname or ip. But because remote debugging works only in a context of local instances (thanks Rob W for pointing this out) you have to, for example, make a tunnel between the port 9222 on your primary computer and another port local to the latter. See this question.