Background: The rapid neutron capture process is one of the main nucleosynthesis processes of elements heavier than Fe. Uncertainties in nuclear properties, such as masses, half-lives, and β-delayed neutron probabilities can cause orders of magnitude of variation within astrophysical r-process simulations. Presently, theoretical models are used to make global predictions of various nuclear properties for the thousands of nuclei required for these simulations, and measurements are required to benchmark these models, especially far from stability. Purpose: β-decay strength distributions can be used to not only inform astrophysical r-process simulations, but also to provide a stringent test for theoretical calculations. The aim of this work is to provide accurate strength distributions for Co69,71β decay. Method: The technique of total absorption spectroscopy was used to measure the β decay of Co69,71 for the first time at the National Superconducting Cyclotron Laboratory. The ions were implanted in a double-sided silicon strip detector at the center of the Summing NaI(Tl) detector and identified using standard particle identification methods. The response of the detection system to the β-decay electron and subsequent γ-ray radiation was fit to the observed experimental data using a χ2-minimization technique. Results: β-feeding intensities and Gamow-Teller strength distributions were extracted from the fits of the experimental data. The β-decay intensities show that there is a large percentage of feeding to levels above 2 MeV, which have not been observed in previous studies. The resultant β-feeding intensities and Gamow-Teller strength distributions were compared to shell model and quasiparticle random phase approximation (QRPA) calculations. Conclusions: Comparing experimentally determined β-decay strength distributions provides a test of models, which are commonly used for global β-decay properties for astrophysical calculations. This work highlights the importance of performing detailed comparisons of models to experimental data, particularly far from stability and as close to the r-process path as possible.