I don’t know much about alternative widgets, but currently with the official widgets you can create on https://twitter.com/settings/widgets, the only way to have a different query per page would be to create a different widget per page. Not a problem if you have 10 pages or less but if you have plenty of them, or the query is dynamic, you will have to find an other way.
You could make your own widgets using the REST and Streaming APIs but that would require quite some work.