Based on the 2 questions that you asked, here is what I would suggest.
First, use unique accounts for each of your users. The person who uses ComputerB should log on to the desktop as UserB. Or use their first name, or function. Any name that uniquely identifies them. The person who uses ComputerC should log on to the desktop as UserC. Again, any account name that is different from what the other user/computer uses.
Then on ComputerA where your Data share is, define local users named UserB and UserC that have the same passwords as the accounts on their respective ComputerB/C.
On the Data folder share permissions, grant "everyone full control". The share permissions act as a filter. That will allow update requests to get through the share.
On the folder permissions, grant the access that you want for each subfolder.
C:\Data should have "Administrators full control", and your account with full control.
C:\Data\ComputerB add UserB full control.
C:\Data\ComputerC add UserC full control.
You will also need to grant read access to C:\Data for your users so that they can "see" the subfolders. But you don't want that permission to be inherited by the subfolders because would allow read access to the other computer's data. You will need to click on the "Advanced" button, and add "Authenticated Users" read access, but in "Applies to" you need to select "This folder and files".
It should look like this.
That will allows the uses to authenticate to ComputerA and the Data share folders will restrict access. You users will not be prompted to type in an id/password when they try to access \\ComputerA\Data. Because the respective passwords match, ComputerA will "know" who the user is.