I found an easy resolution for this issue was to add
source /etc/profile
to the top of the script.sh file I was trying to run on the target system.
On the systems here, this caused the environmental variables which were needed by script.sh to be configured as if running from a login shell.
In one of the prior responses it was suggested that ~/.bashr_profile etc... be used.
I didn't spend much time on this but, the problem with this is if you ssh to a different user on the target system than the shell on the source system from which you log in it appeared to me that this causes the source system user name to be used for the ~.
