In this work, we present COSTREAM, a novel learned cost model for Distributed Stream Processing Systems that provides accurate predictions of the execution costs of a streaming query in an edge-cloud environment. The cost model can be used to find an initial placement of operators across heterogeneous hardware, which is particularly important in these environments. In our evaluation, we demonstrate that COSTREAM can produce highly accurate cost estimates for the initial operator placement and even generalize to unseen placements, queries, and hardware. When using COSTREAM to optimize the placements of streaming operators, a median speed-up of around 21x can be achieved compared to baselines.