The "Any Local Repository" in the permissions page affects remote repositories. Seems like it affect remotes repos only when the "Store Artifacts Locally" is disabled
Steps to reproduce:
1. Create a new remote repository ( for example jcenter)
2. Disable "Store Artifacts Locally" in the advanced tab.
3. Create a new user, make sure he is in the readers group and that the readers group included the "anything" permissions target.
4. Uncheck only the "Any Local Repository" in the anything permission target.
5. Try to pull a package from the remote repository, in case of jcenter:
$ curl https://<art-url>/artifactory/jcenter/5.1.4/connector-java-5.1.4-bin.jar -uuser:pass
6. You will get 403 although we should not as we have "Any Remote Repository" enabled in the anything permission target..
7. Enable "Any Local Repository", you will be able to pull the package as expected.