Transactional memory promises to make concurrent programming tractable and efficient by allowing the user to assemble sequences of actions in atomic transactions with all-or-nothing semantics. It is believed that, by its very virtue, transactional memory must ensure that all committed transactions constitute a serial execution respecting the real-time order. In contrast, aborted or incomplete transactions should not "take effect." But what does "not taking effect" mean exactly? It seems natural to expect that aborted or incomplete transactions do not appear in the global serial execution, and, thus, no committed transaction can be affected by them. We investigate another, less obvious, feature of "not taking effect" called non-interference: aborted or incomplete transactions should not force any other transaction to abort. In the strongest form of non-interference that we explore in this paper, by removing a subset of aborted or incomplete transactions from the history, we should not be able to turn an aborted transaction into a committed one without violating the correctness criterion. We show that non-interference is, in a strict sense, not implementable with respect to the popular criterion of opacity that requires all transactions (be they committed, aborted or incomplete) to witness the same global serial execution. In contrast, when we only require local correctness, non-interference is implementable. Informally, a correctness criterion is local if it only requires that every transaction can be serialized along with (a subset of) the transactions committed before its last event (aborted or incomplete transactions ignored). We give a few examples of local correctness properties, including the recently proposed criterion of virtual world consistency, and present a simple though efficient implementation that satisfies non-interference and local opacity.