1. A system, comprising one or more computing devices of a provider network operable to implement a network-accessible service for generating random data, wherein the one or more computing devices are configured to: implement one or more programmatic interfaces enabling a determination of respective characteristics of random data to be provided to one or more clients of the service; implement one or more security protocols for transmission of random data to the one or more clients, the one or more security protocols including at least one security protocol for transmission of random data to trusted clients at one or more devices resident within the provider network; and in accordance with a determined set of characteristics of random data to be provided to a particular client of the one or more clients: obtain, on behalf of the particular client, random data from one or more servers of the provider network designated as random data producers for the one or more clients; and initiate a transmission of the random data obtained from the one or more servers, directed to a destination associated with the particular client, in accordance with a security protocol determined for the transmission.