You can’t use Business rule to open up new screen while you are in one screen. One option you can do is, create hyperlink on any visible field (ex. request Id field) in person screen and click that hyperlink value to open request screen. (this happened automatically once you set proper hyperlink, for ex, you can open contract screen from Request screen through contract hyperlink in request screen.) but this is not something dynamic as you always need to manual work to open another screen.
When it consider Script, all the function related to screen are Mobile client related, so that you cant expect that it works in back office.(smart client)
Other way round, This looks like Client Side customization, where you can trigger any custom screen/widget by calling custom client dll over any custom client event. (you would need to go through technical training to do this!)
In the Web Client, this can be done with client scripting. You can also do this in the Smart Client by setting up lookup navigation e.g. a ‘breadcrumb’ for a button via the UI Designer. The metadata which defines the navigation item for a button in the Smart Client is ignored by the Web Client, so you need to use one approach and/or the other depending upon which platform(s) you need to support if you want use a button to navigate to another screen.