Motivated by applications in distributed storage and distributed computation, we introduce embedded index coding (EIC). EIC is a type of distributed index coding in which nodes in a distributed system act as both broadcast senders and receivers of information. We show how linear embedded index coding is related to linear index coding in general, and give characterizations and bounds on the communication costs of optimal embedded index codes. We also define task-based EIC, in which there is only one sender node responsible for transmitting a block to a particular receiving node. Task-based EIC is more computationally tractable and has advantages in applications such as distributed storage, in which senders may complete their broadcasts at different times. Finally, we give heuristic algorithms for approximating optimal linear embedded index codes, and demonstrate empirically that these algorithms perform well.