A fine point:
When setting Windows permissions on the .MDB file and the folder in which it resides, ...
When you right-click on the object to get properties, then follow that to the security settings tab, you see the "basic" permissions. You need MODIFY access to the folder. You do not need FULL CONTROL.
If you go "Advanced" from there, you need every permission EXCEPT "Take Ownership" and "Change Permissions." I don't think it matters whether you have "Read Permissions."
Don't forget that if your shared folder isn't a first-level folder, you will also need to have "Traverse Folder" on all parents and "List Folder" on the folder holding your DB. (You will need "Traverse Folder" on the MFD but usually this is allowed by default anyway.)
If your application involves child folders that Access will also travers, then you need to set up the folder first, then use Windows inheritance options to propagate the permissions to the child files and folders.
You MUST have "Create Files" and "Delete" on the folder contents. Your network security officer might wish to prevent you from having "Delete" on the folders themselves, but inheritance can be made to do this correctly.
Symptom: MDB comes up as though opened exclusively. Cause: You cannot create the .LDB file in the appropriate place, so you cannot share lock data. In that case, Access has no choice but to open the MDB exclusively.
If you aren't sure about what I just told you, get with your sys admin or network admin. They should understand what I'm saying here.