No, it depends on:
- the permissions of the mounted filesystem (if the filesystem has the noexec option set, nothing on that filesystem will run)
- the permissions and owner of the file. A file can be marked as executable for the owner user, the owning group, or everyone. By default, most packages set themselves to be run by everyone. (NB: scripts also have to be readable before they can be run)
- the permissions of the directory that the file is in. If a user cannot read the directory, then they can't run executables in that directory.
Security packages like sudo can be used to add more fine-grain tuning to who can use what.