Distributed function computation is the problem, for a networked system of $n$ autonomous agents, to collectively compute the value $f(v_1, \ldots, v_n)$ of some input values, each initially private to one agent in the network. Here, we study and organize results pertaining to distributed function computation in anonymous networks, both for the static and the dynamic case, under a communication model of directed and synchronous message exchanges, but with varying assumptions in the degree of awareness or control that a single agent has over its outneighbors. Our main argument is three-fold. First, in the "blind broadcast" model, where in each round an agent merely casts out a unique message without any knowledge or control over its addressees, the computable functions are those that only depend on the set of the input values, but not on their multiplicities or relative frequencies in the input. Second, in contrast, when we assume either that a) in each round, the agents know how many outneighbors they have; b) all communications links in the network are bidirectional; or c) the agents may address each of their outneighbors individually, then the set of computable functions grows to contain all functions that depend on the relative frequencies of each value in the input - such as the average - but not on their multiplicities - thus, not the sum. Third, however, if one or several agents are distinguished as leaders, or if the cardinality of the network is known, then under any of the above three assumptions it becomes possible to recover the complete multiset of the input values, and thus compute any function of the distributed input as long as it is invariant under permutation of its arguments. In the case of dynamic networks, we also discuss the impact of multiple connectivity assumptions.