Analytics tracks your visitors… if there’s a referrer passed to Google it tracks that as well and in the case of social services it even breaks down the detail into a special reporting area… If they skipped Twitter they’d have had a mass exodus.
While twitter’s official shortener is the most common it is not the only source and in some cases links are entirely masked or provide no referral history so while the visit is tracked, you won’t know where they came from.
As for tracking their ad clicks, do you filter by source for other traffic? If so it’s the same methodology, keeping in mind my comments from above.
I’m pretty sure that they can still track that traffic coming regardless of where it is coming from. I think that’s the special part about analytics, is that you can really study up your site with a simple tool.