[Rediscover]

Probably not the answer You are looking for...

Try some variant of this in your remote .bash_profile or what-not:

  if [ "$PS1" ]; then
      export PS1='\h:\w\$ '
      if [[ "x${TERM}" = "xeterm" || "x${TERM}" = "xeterm-color" ]]; then
       function set-eterm-dir {
         echo -e "\033AnSiTu" $(whoami)
         echo -e "\033AnSiTc" $(pwd)
         echo -e "\033AnSiTh" $(hostname -f)
       }
      PROMPT_COMMAND=set-eterm-dir
    fi
  fi

Now, locally, invoke M-x ansi-term, then within the ansi-term ssh to the remote machine. Change dirs, hit C-x C-f to open a file in the current (remote) directory.

See also the ansi-term hints on emacswiki

https://www.emacswiki.org/emacs/AnsiTermHints#h5o-5

or the comments in /usr/local/share/emacs/*/lisp/term.el

Edit: formatting

[Rediscover]

Apologies, I meant to add more saying I could not get a remote machine to invoke the local emacs reliably. I got close with hacking up emacsclient to ssh back, but was eventually stymied by new network policies at $JOB and eventually settled for the above.

db48x has a great answer below.