Context. Nuclear regions of ultraluminous and luminous infrared galaxies (U/LIRGs) are powered by starbursts and/or active galactic nuclei (AGNs). These regions are often obscured by extremely high columns of gas and dust. Molecular lines in the submillimeter windows have the potential to determine the physical conditions of these compact obscured nuclei (CONs). Aims. We aim to reveal the distributions of HCN and HCO+ emission in local U/LIRGs and investigate whether and how they are related to galaxy properties. Methods. Using the Atacama Large Millimeter/submillimeter Array (ALMA), we have conducted sensitive observations of the HCN J = 3-2 and HCO(+)J = 3-2 lines toward 23 U/LIRGs in the local Universe (z < 0.07) with a spatial resolution of similar to 0.3 '' (similar to 50-400 pc). Results. We detected both HCN and HCO+ in 21 galaxies, only HCN in one galaxy, and neither in one galaxy. The global HCN/HCO+ line ratios, averaged over scales of similar to 0.5-4 kpc, range from 0.4 to 2.3, with an unweighted mean of 1.1. These line ratios appear to have no systematic trend with bolometric AGN luminosity or star formation rate. The line ratio varies with position and velocity within each galaxy, with an average interquartile range of 0.38 on a spaxel-by-spaxel basis. In eight out of ten galaxies known to have outflows and/or inflows, we found spatially and kinematically symmetric structures of high line ratios. These structures appear as a collimated bicone in two galaxies and as a thin spherical shell in six galaxies. Conclusions. Non-LTE analysis suggests that the high HCN/HCO+ line ratio in outflows is predominantly influenced by the abundance ratio. Chemical model calculations indicate that the enhancement of HCN abundance in outflows is likely due to high-temperature chemistry triggered by shock heating. These results imply that the HCN/HCO+ line ratio can aid in identifying the outflow geometry when the shock velocity of the outflows is sufficiently high to heat the gas.