Not the OP but I tried to translate this site out of curiosity.
Conditions/Test Steps
1. I am already authenticated and in an active session.
2. From the Google Translate page, I entered the URL for this thread to get a translation from English to Italian.
2. I am able to enter the code, and land on the authentication page, but it is in English, not Italian.
3. If I click the Translate button in the Google Translate header at the top of the page whatever cookie is used to indicate I've entered my secret beta code is no longer valid and I am kicked back to the secret code page.
4. Even if I wait until I've fully authenticated before trying again to translate a page using the Translate button,, the result is the same as described in the previous step.
I'm not sure if this is due only to the additional gateway (i.e. secret beta code) or how authentication works on the site in general. The normal behavior on other sites I visit is that navigation from page to page doesn't default me to the native language of the website. The pages are displayed in the language I have selected in the drop down as the "translate to" language.