How do I set the private file system path's permissions? There is an option in Configuration > File system which let you  define a path for private files. After setting a path in this field, the access for all users is restricted (including admin). I need to give the permission of accessing this directory and its files to some users. How?
Your module has to implement hook_file_download(). By default all private files will give access denied. 

If your access rules are not complex, you can also use one of contrib modules, ex. Private files download permission