Christina, this is a bit tricky situation as it involves cross-domain tracking and the visitor/goal tracking won't be consistent since each site is using its own sets of cookies. To keep this not that complicated and technical (for that purpose, there is a lengthy documentation by Google), I am going to assume that the conversion will be counted as simply the fact that the user pressed button.
You could then setup event tracking and create a new goal with a condition of this event being fired (Type = Event, in Goal details then use the same Category/Action/Label as you would you in the tracking code -- see below).
The example of event tracking code: (see the complete specification for details)
<a href="#" onClick="_gaq.push(['_trackEvent', 'Goal', 'Submit', 'Name']);">Submit</a>
Hope this helps :)