Backup database on linked server
You wrote that &quot;I have given NT SERVICE\MSSQLSERVER full control&quot;, so it looks like your SQL Server is running under &quot;local system accounts&quot;.
If you granted full control to NT SERVICE\MSSQLSERVER on a second server, it will not work, because NT SERVICE\MSSQLSERVER is a different account on each machine.
You need to grant all network privileges to account [DomainName]\[ServerName]$ for a local service accounts to have access to network resources.
In your case it will be [DomainName]\ServerA$ account and this account will need to have access to the network share.