Additional SQL Server features and topics not covered by specific categories
Your question is not clear, but SSMS sure can read user folders. Just select the Open or Save menus to verify. SSMS runs with your permission, so anything you can access, SSMS can access.
However, if you are browsing for the location of a backup or a place to store a database file it is a different matter. In this case, SSMS asks SQL Server to browse the file system. This is quite natural, since these are folders to be used in SQL commands. I guess you have SQL Server running on your local machine, but keep in mind that this is a special case. Normally, SQL Server runs on different server than your workstation.
And SQL Server does not normally have access to the other user's folders. Or, rather, SQL Server runs under a service account, and that service account has access to its own user folder, but not yours.