Every other person in the world who has a twitter page is responsible for the content that gets put up on it. Politicians are no exception to this rule. Although politicians often have someone paid to put information up, they are still responsible for hiring the individual who puts it up.
Unless a politician can plausibly say that they have had their account hacked or otherwise maliciously accessed, the politician should be held responsible for its content. Twitter is a platform as valid as a press release or a newsletter, and any correspondence from the account is a representation of official views.
Yes, politicians are responsible for the content on their twitter pages. Even though it is unlikely that they maintain the account themselves and do not give approval for each tweet, they should have a clear enough leadership style that their staff knows what is appropriate and what isn't.
While in some cases a tweet may not reflect the politician at all, they are still responsible for apologizing and addressing any fallout, including disciplining staffers.