I'm Kasra Rahjerdi, a Software Engineer at Fitocracy. Before I joined the team, I actually created a reverse engineered copy of the mobile site's API with the goal of creating my own Android app off of it, so I can relate to what you're asking.
To directly respond to your question: Even if an app is open-source, the average user is going to be confused and hesitant to hand over their data to a third party, unless the developer name for the application is the company's name. I assume you're asking this question in regards for creating an API to create an Android application, in which case I'd strongly recommend against doing this because publicly releasing an application and API based off of our information will lead to hassles for the both of us. Users finding your application will be contacting you asking you why X doesn't work where it works on the iPhone app, and users finding the product through you will be leaving our ecosystem after the initial impression because the #1 thing our iPhone app is based off of is the perfect UX that we want.
But, I can't stop you from doing anything, so here's a GitHub project of another reverse engineered version of our API with a barebones Android app that you can use as a stepping stone if you want: https://github.com/walialu/Unofficial-Fitocracy.com-Android-App I'll just tell you that the creation of that application created a lot of chaos for us, since we had confused users left and right.
And to go back to your direct question: In the day and age of oAuth I really believe that the average consumer is hesitant to give user information unless they're on the application website of the supplier of data, so I wouldn't ask the user for username/password.