Is it important that the folder be shared with one specific person, or can it be shared with everyone? I believe that to restrict it to one particular person he/she must have an account (and associated password) on "your" (the sharing) mac.
Go to System Preferences-->Sharing. On the left side under "service", select File Sharing and turn it on. In the Shared Folders box, by default, you'll see the Public folders of every user account on that Mac. Click the '+' below that box and you'll be able to select one of your folders to be added to the shared folders.
Making sure that the newly-added folder is currently selected in the "Shared Folders" box, look at the "Users" box. It appears that by default You will have read/write access, and Everyone will have read-only access to it. You can change Everyone's access to "No Access" if you like. Use the '+' below the "Users" box to add access for the person you want. It displays a list of users & groups on your Mac. If you select a user from this list, I believe they will have to enter that user's password to connect to the shared folder (and no one else should be able to). Now, still in the "Users" box, set the access for that user to Read Only or Read-Write, as you wish.
Interestingly, that dialog box also shows Contacts. If you choose one of them, I have no idea how macOs verifies/authenticates that person. You could experiment.
Now, on another Mac on the same network, the authorized user can use Finder, select your Mac on the left side, and use the Connect As... button to gain access to (only) the shared folder(s) (entering the username and password of the account on "your" Mac). Of course, if it's important I would verify that only the shared folder(s) show in the list he/she has access to.
Hope this helps!