Device arbitration by multiple speech processing systems

A device can perform device arbitration, even when the device is unable to communicate with a remote system over a wide area network (e.g., the Internet). Upon detecting a wakeword in an utterance, the device can wait a period of time for data to arrive at the device, which, if received, indicates to the device that another speech interface device in the environment detected an utterance. If the device receives data prior to the period of time lapsing, the device can determine the earliest-occurring wakeword based on multiple wakeword occurrence times, and may designate whichever device that detected the wakeword first as the designated device to perform an action with respect to the user speech. To account for differences in sound capture latency between speech interface devices, a pre-calculated time offset value can be applied to wakeword occurrence time(s) during device arbitration.