Miscellaneous general commentary:

• You shouldn't use /^\.\.?$/ to throw away the special directories. $ will disregard a trailing newline, so you'll trip over files named "..\n" and such. The proper pattern would be /\A\.\.?\z/, but two humble eqs will do just as well ($_ ne '.' and $_ ne '..').
• As already mentioned, the file tests and everything else works off of the current directory, so you need to prepend any path you want to them to be relative to to the filename.
• You're looping over the filename list three times: once in the grep, then twice in one additional foreach loop per test. A lot of duplicated work.
• You first throw away the special directories, then every other entry that starts with a dot. Since both special directories start with a dot, why not pull these tests together?

Try something like this:

#!/usr/bin/perl -w
use File::Spec::Functions qw(catfile);
use strict;

my @users = qw(/home/zentara);

foreach $homedir (@users) {
    my $dh;
    opendir $dh, $homedir
        or warn "Cannot readdir $homedir:$!\n";

    my @files =
        grep /\A[^.]/ and -f catfile($homedir, $_),
        readdir DIR;

    print "@files\n" . "#"x47 ." \n";
}
exit;
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File::Spec::Functions provides importable versions of File::Spec's crossplatform functions for de/constructing pathnames. These modules are in the Perl core.

Update: s/readdir DIR/readdir $dh/

Makeshifts last the longest.