I'm looking for a way to search a directory or directories and list all the files that have the wrong permissions for a public directory.
Everything depends on what do you consider 'incorrect permission'. man find helps you by defining the way how you can look for files/dirs with given permission:
-perm -mode All of the permission bits mode are set for the file. Symbolic modes are accepted in this form, and this is usually the way in which would want to use them. You must specify ‘u’, ‘g’ or ‘o’ if you use a symbolic mode. See the EXAMPLES section for some illustrative examples. -perm /mode Any of the permission bits mode are set for the file. Symbolic modes are accepted in this form. You must specify ‘u’, ‘g’ or ‘o’ if you use a symbolic mode. See the EXAMPLES section for some illustrative examples. If no permission bits in mode are set, this test matches any file (the idea here is to be consistent with the behaviour of -perm -000). -perm +mode Deprecated, old way of searching for files with any of the permission bits in mode set. You should use -perm /mode instead. Trying to use the ‘+’ syntax with symbolic modes will yield surprising results. For exam‐ ple, ‘+u+x’ is a valid symbolic mode (equivalent to +u,+x, i.e. 0111) and will therefore not be evaluated as -perm +mode but instead as the exact mode specifier -perm mode and so it matches files with exact permissions 0111 instead of files with any execute bit set. If you found this para‐ graph confusing, you’re not alone - just use -perm /mode. This form of the -perm test is deprecated because the POSIX specification requires the interpretation of a leading ‘+’ as being part of a symbolic mode, and so we switched to using ‘/’ instead.