Compressive sensing (CS) has been shown useful for reducing dimensionality, by exploiting signal sparsity inherent to specific domain representations of data. Traditional CS approaches represent the signal as a sparse linear combination of basis vectors from a prescribed dictionary. However, it is often impractical to presume accurate knowledge of the basis, which motivates data-driven dictionary learning. Moreover, in large-scale settings one may only afford to acquire quantized measurements, which may arrive sequentially in a streaming fashion. The present paper jointly learns the sparse signal representation and the unknown dictionary when only binary streaming measurements with possible misses are available. To this end, a novel efficient online estimator with closedform sequential updates is put forth to recover the sparse representation, while refining the dictionary 'on the fly'. Numerical tests on simulated and real data corroborate the efficacy of the novel approach.