Is the user logged in at the point they click the Edit button? If so then the user's session information should be part of the redirect and therefore they wouldn't get a login prompt. It appears that the authentication information isn't being passed along. You can watch the browser's network traffic to confirm that the session information is getting passed along properly.
Given you used the web forms tag I'm assuming this is a web forms app correct?
Can you discuss the logic for your edit button? I assume it is an actual button. I also assume you have a server side event handler reacting to the button click. It does whatever and then calls the redirect. For your specific redirect to work the .aspx page needs to be in the same directory as the current page, otherwise you need to use either a virtual path or resolve the full URL first.
Yes Redirect works locally. What Important Information I should not remove?