Are there any issues with installing memcache on a VPS running multiple Drupal sites?
Make sure your settings.php file has a unique prefix for each site that has memcache enabled.  I got tripped up by this, and everything went berserk. This thread fixed it for me.  Basically, this is the only issue you would have as far as conflicts.

http://drupal.org/node/1416102

Like APC, memcache works indepedently of your sites.  Make sure you have enough memory for it to function correctly.  You can get a rough idea idea by seeing how much disk space all of your drupal databases take up.  Not all tables in each database will be cached of course, but so long as you have that much memory to dedicate to memcache, you are good to go.

Said differently, as an example I have 12 sites on my VPS 4 gigs of RAM total with 1 Gig dedicated to memcache.  I have never had a problem. Total size of databases on disk is around 400 mb.