There are two things to watch out for here.
First, you have to be showing both hidden and system files to see the contents of
$Recycle.Bin. Open your Folder Options dialog and switch to the View tab. Select Show hidden files, folders, and drives and uncheck Hide protected operating system files.
Then, you'll be able to see the subfolders of
$Recycle.Bin in Explorer. The only subfolder you'll be able to access is the one corresponding to the user you're authenticated as; that folder will show up as Recycle Bin with a recycle bin icon. (The others show up as SIDs.) But when you open your subfolder, you'll see the contents of the Recycle Bin of the machine you're browsing from. That's because the
desktop.ini in that folder sets a special shell folder view for it, and Explorer thinks it should display the current user's Recycle Bin contents from the current machine.
You could use the command prompt to poke around that folder. If you really want to use Explorer, you need to make it so the Explorer you're browsing from can't process
desktop.ini. On the machine that owns the Recycle Bin in question, use a command prompt to navigate to the subfolder your user owns. Use trial-and-error to get the right one (
dir /a and tab completion will help here) or look up your SID with
whoami /all. Once you're in, run this command to block only network access to
icacls desktop.ini /deny NETWORK:F
You'll then be able to browse that folder over the network like any other, and local use of the Recycle Bin will be unaffected. Unfortunately, the file names will be garbled because of how the Recycle Bin stores things. You'll see the folder contents as they really are on-disk.
If you want to undo that
icacls change, use this:
icacls desktop.ini /remove NETWORK