rsh server "script..." which $PATH is used ?

When running a script on a distant server via rsh, what is the $PATH used ?

I had done a script in the /usr/local/bin but the rsh reported it did not find it. So I assume it is using a separate $PATH but how can I find out ?

rsh requires fully qualified names of executables, scripts to be safe. It uses /usr/bin/sh (whatever that evaluates to on the remote server), and the PATH is usually limited:
/usr/bin:/usr/sbin

rsh remotebox -l myusername "/usr/bin/echo my path is \$PATH"

might help.

You can do this to set a profile

rsh remotebox -l myusername ". .profile  && /usr/bin/echo my path is \$PATH"