start up emacs using one of those remote directories as if it were local
If the servers are unix/linux servers and you have ssh access, then you could try SSH Filesystem. Then you can mount the servers to local directories, e.g.:
> sshfs server1: ~/remote/server1
> sshfs server2: ~/remote/server2
Then you could start emacs with emacs --no-init-file --load ~/remote/servers2/.emacs
and so on.
packages that, for various reasons, I don't want to store locally
If packages are installed in .emacs.d on the remote machines you could create scripts like the following on your local machine:
;; .emacs.server1.el
(add-to-list 'load-path (expand-file-name "~/remote/server1/.emacs.d"))
(add-to-list 'load-path (expand-file-name "~/remote/server1/.emacs.d/package1"))
(load (expand-file-name "~/remote/server1/.emacs"))
And then start emacs like this: emacs --no-init-file --load ~/.emacs.server1.el
Obvious this script depends on the mounts above.