I have shared hosting at Hot Drupal with SSH access. I have a couple of sites in subdirectories of my HTML root.
This is what worked for me.
I could not install Drush to /usr/bin/lib so I put it in my home directory
Then with PuTTY I navigated to the subdirectory where I want to use Drush. I created an alias with
alias drush="~/whereyouputdrush/drush" where the ~/ is my home directory.
then I was able to use right there eg
firstname.lastname@example.org [~/public_html/meier]# drush help
(this is repeated from http://drupal.org/node/513896#comment-2346908 b/c I think this method should be added to the documentation, which currently says
$ alias drush='/path/to/drush/drush' but not how to set path to your home dir with ~/)