We investigate the complexity of embeddings between bi-embeddable structures. In analogy with categoricity spectra, we define the bi-embeddable categoricity spectrum of a structure $mathcal A$ as the family of Turing degrees that compute embeddings between any computable bi-embeddable copies of $mathcal A$; the degree of bi-embeddable categoricity of $mathcal A$ is the least degree in this spectrum (if it exists). We extend many known results about categoricity spectra to the case of bi-embeddability. In particular, we exhibit structures without degree of bi-embeddable categoricity, and we show that every degree d.c.e. above $mathbf{0}^{(alpha)}$ for $alpha$ a computable successor ordinal and $mathbf{0}^{(lambda)}$ for $lambda$ a computable limit ordinal is a degree of bi-embeddable categoricity. We also give examples of families of degrees that are not bi-embeddable categoricity spectra.
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