Accessing a widget variable from a theme

I have a custom widget in my theme that allows the blog owner to display recent tweets from Twitter. One of the variables within the widget code is $twitter_id. I’d like to be able to access this variable from within the theme page code itself so I can use it in other areas of the page (for example, a button that allows you to link directly to the person’s twitter feed).

I’ve been searching around and haven’t found a good way to do this yet. I feel like I’m probably missing something simple, but would appreciate any help I can get!