As far as I know, there is no option to not save sent messages in Gmail. You probably can create a filter to delete them. Create a filter with your account in the From field, and set it to delete then it will be put in the Trash immediately. This still won’t stop other from trying to read any emails. If that is the intent, I’d set up an account for each person. Then set up a rule if you get an email that more than one person needs to see, creat a rule to forward the new email to those people.