In the question that you posted you have:
$destinationPath1 = 'D:\folder1\'
Where does the D drive point to? Is that a drive letter that you mapped to the destination server? Are you able to view and update the files/folders on that share?
If the user account that the robocopy/powershell processes are running as, does not have access to the \DestinationServer\ShareName then on the NET command you can provide the credentials (userid+password) so that the destination server can authenticate the user. You can also use Powershell's New-PSDrive cmdlet to map a network drive with alternate credentials.
Another trick is to use a temporary local account, maybe named DataMigration, and put it in the Administrators group on both the source and destination servers. Use the same password on both machines. RDP to the source server with that account. You should then be able to access \DestinationServer\ShareName.
Update: Example using New-PSDrive.
$User = ".\admin"
$PWord = ConvertTo-SecureString -String "admin" -AsPlainText -Force
$Credential = New-Object -TypeName System.Management.Automation.PSCredential -ArgumentList $User, $PWord
New-PSDrive -Name "N" -PSProvider "FileSystem" -root "\\test10\temp" -credential $Credential
$src = 'c:\data'
$dst = 'n:\data'
robocopy $src $dst /l
Remove-PSDrive -Name "N"