For a function with strict saddle points, we show that the classical trust-region algorithm with exact subproblem minimization finds approximate (local) minimizers in polynomial-time. The accuracy parameter $\varepsilon$ only appears in a $\log\log(\varepsilon^{-1})$ term. In that sense, the complexity is essentially independent of the accuracy $\varepsilon$: using a second-order model allows to benefit from the quadratic convergence rate of Newton's method near minimizers. This is a notable improvement over known results for generic nonconvex functions, where the worst-case number of iterations is of the order $\mathcal{O}(\varepsilon^{-3})$. Our complexity bound depends polynomially on landscape parameters which represent negative curvature and strong convexity constants. In the context where these landscape parameters are known, we present a strict saddle trust-region method with similar complexity guarantees which minimizes the trust-region subproblem approximately. It is based on a truncated conjugate gradient scheme and uses the Lanczos method to compute directions of negative curvature. The complexity result gives an upper bound on the total number of gradient evaluations and Hessian-vector products. All our results apply to the setting where the function is defined on a Riemannian manifold, and Riemannian algorithms are used