There are several methods. One is to disconnect from Twitter (with which you log in at Twitpic). See below how to revoke access. This means it’s no longer possible to login using your Twitter account at Twitpic (this can be done again afterwards if you would wish to). Two other methods for account deletion can be clearly read here: https://twitpic.helpmint.com/articles/how-do-i-delete-my-twitpic-account
How about data disclosure by Twitpic? The ToS state “You retain all ownership rights to Content uploaded to Twitpic. However, by submitting Content to Twitpic, you hereby grant Twitpic a worldwide, non-exclusive, royalty-free, sublicenseable and transferable license to use, reproduce, distribute, prepare derivative works of, display, and perform the Content in connection with the Service and Twitpic’s (and its successors’ and affiliates’) business, including without limitation for promoting and redistributing part or all of the Service (and derivative works thereof) in any media formats and through any media channels.” We are no experts in law, but it seems to mean: they can do nearly whatever they like, as long as they don’t earn money directly through your submissions.
How about data retention? The ToS state “Twitpic may retain, but not display, distribute, or perform, server copies of your media that have been removed or deleted. The above licenses granted by you in user comments you submit are perpetual and irrevocable. Deleted images are only accessed in the event of a legal issue.” This is pretty standard. So, even after some ‘public’ deletion, your data may linger around for a while.