The 'Loadsheet' can mean different things to different airlines. I work for a commercial airline in the US, and to us a 'Loadsheet' is what other airlines call a 'Load Report'. If you are referring to a 'Trim Sheet' where the live and deadloads loads below and above the wing are converted in to flight data (such as ZFW, TOW, MAC, StabTO etc) then most large airlines will transmit that data direct to the cockpit via ACARS using Automatic Load Planning Software, once the flight closes and all bags / cargo / mail has been loaded and reported.
Back when I was a dispatcher in the UK, we either used a Load Control program to input the weights and print off the sheet for the flight crew, or resort to the old manual style load sheet.
Technology has moved on now so most Trim Sheets are sent direct to the flight crew. These are obviously stored digitally so having a copy 'On The Ground' is not usually required in the traditional sense.