A speech interface device is configured with “hybrid” capabilities, which allows the speech interface device to perform actions in response to user speech, even when the speech interface device is unable to communicate with a remote system over a wide area network (e.g., the Internet). A hybrid request selector of the speech interface device sends audio data representing user speech to both a remote speech processing system and a local speech processing component executing on the speech interface device, and then waits for a response from either or both components. The local speech processing component may start execution based on the audio data and subsequently suspend the execution until further instruction from the hybrid request selector. The hybrid request selector can then determine which response to use, and, depending on which response is chosen, may instruct the local speech processing component to either continue or terminate the suspended execution.