The following uses exclusively cocycle descriptions for spin and spinc structures which I would like to avoid. See for example Nicolaescu "Notes on Seiberg-Witten invariants", pages 40-41 for their use in spin geometry. I know how to define spinc structures via principal bundles which I consider more concise.
Let $M$ be an oriented smooth $n$-dimensional Riemannian manifold. Let $\rho:\operatorname{Spin}(n)\to \operatorname{Aut}(\Delta_n)$ be the complex spinor representation and $\rho^\mathrm{c}:\operatorname{Spin^c}(n)\to \operatorname{Aut}(\Delta_n)$ be the "spinc version" of the complex spinor representation.
The definition of a spinc structure needs a $\operatorname{SO}(n)$-principal bundle or equivalently a Riemannian metric. We can describe it via cocycles for an open cover by charts $\mathfrak{U}=(U_i)_{i\in I}$. Denote $U_{ij}:=U_i\cap U_j$.
The manifold $M$ is said to possess a spinc structure if there exist smooth maps $g^\mathrm{c}_{ij}:U_{ij}\to \operatorname{Spin^c}(n)$, satisfying the cocycle condition and such that $\rho^{\mathrm{c}}(g^\mathrm{c}_{ij})=g_{ij}$.
Because of $\operatorname{Spin^c}(n)=\operatorname{Spin}(n)\times_{\mathbb{Z}_2}S^1$ this existence is equivalent to the existence of cocycles
$h_{ij}:U_{ij}\to\operatorname{Spin}(n)$ and $z_{ij}:U_{ij}\to S^1$
such that $\rho(h_{ij})=g_{ij}$ and $(h_{ij}h_{jk}h_{ki},z_{ij}z_{jk}z_{ki})\in\{(-1,-1),(1,1) \}$.
Given a spin structure $\mathfrak{s}$ described by cocycles $(h_{ij},z_{ij})$, the spinc structure $-\mathfrak{s}$, also written $\bar{\mathfrak{s}}$, is given by the cocycles $(h_{ij},\bar{z}_{ij})$.
So how can I formulate this definition in terms of spinc structures as spinc principal bundles?