Going directly through a local owner has its upsides, the prices are usually lower and people tend to be more flexible with pick-up times as long as you communicate properly. The flip side is you really need to check the car thoroughly because maintenance records are hit or miss and you do not want to be stranded on a mountain pass. Finding a car to rent across Kyrgyzstan through private listings is pretty easy on sites like this one
https://getrentacar.com/en-US/kyrgyzstan .