Description of the issue
Insync may sometimes fail to download files or folders from Drive with the error message "Can't process [file/folder name] - Permission denied".
These errors are most commonly due to Insync not having write permissions to the syncing location.
Verifying the cause of the issue
You can verify whether that is the cause by going to the containing folder in Finder, right-clicking it, and clicking Get Info.
In this example, the user jomel only has read-only permissions.
Steps to add read & write permissions to a folder
1. In Finder, navigate to the folder you want to change the permissions for. You can either modify permissions just for the particular folder encountering the error, or choose to add read & write permissions to your whole syncing folder. Right-click the folder and click "Get Info". This will open the dialog pictured above.
2. Click the lock icon on the bottom-right. This will prompt you for your password. Type your password and click OK.
If you're not an administrator on the machine, you may see this dialog instead:
In this case, either provide the username and password of an administrator account, or ask an administrator to perform the permission change.
3. Click the "Read only" privilege corresponding to your user, then select "Read & Write" from the options that show up.
4. Click the gear icon, then click "Apply to enclosed items..."
5. In the Insync window, either click "Retry" on each of the errors, or just click "Retry all".