Find out what printers a user has mapped remotely

Well, you can take a look at the Win32_Printer WMI class,

Get-WMIObject Win32_Printer -ComputerName $Comp

But, I think this will get you even better results:

New-PSSession $Comp | Enter-PSSession
Get-ChildItem Registry::HKEY_Users$UserSIDPrintersConnections

Without using PS Remoting, you could do this instead:

$Printers = [Microsoft.Win32.RegistryKey]::OpenRemoteBaseKey(Microsoft.Win32.RegistryHive]::Users, $ServerName)

You get the idea. Basically, you need to access that user’s registry key with whatever method and enumerate whatever you find in PrintersConnections.