EDITED: November 27, 2014
Twitter recently released Fabric that does include SSO support. From the Android docs:
When attempting to obtain an authentication token, the Kit will use the locally installed Twitter app to offer a single sign-on experience. If the Kit is unable to access the authentication token through the Twitter app, it falls back to using a web view to finish the OAuth process.
My older response is below.
tl;dr
Currently, SSO with Twitter is not possible on Android.
Longer version:
What you are basically asking for is Single sign-on (SSO), similar to what is available with the Facebook SDK for Android. Twitter has never bothered to even implement their own library for Android, let alone SSO. There are a bunch of people who have asked on the dev forums (here and here) but the standard response is, and I quote,
We do not currently offer a SDK for Android with the same level of tight integration as is available in iOS5.
Twitter needs to first get their Android app to support SSO. I would suggest using Facebook and Google+ for login instead of Twitter.
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