Hello jaguargene ...
Klondike here.....
I built one ... maybe 30 plus years or so ago... and there are a number of good ways to drive the drum... below is the way I decided on...
I used a rubber, toothed timing belt..... screwing one belt to the trommel barrel inside out (Teeth up) and an second exact same belt to fit on top of that belt (teeth down).. and to a 1 1/2 to 3 inch toothed timing pulley with the same tooth pattern as the belts for a good fit..... The pulley was either directly mounted to the B & S engine output shaft.... I ran it several years before needing to replace either one of the belts... With this arrangement... no matter how wet the belt gets, it won't slip..it always had good drive.. tooth belt mounted against tooth belt.....and the belts are cheaper than chain, they won't stretch any more or less than a chain will stretch... and the belts are more easily mounted to the drum than chain....and will take about 20 minutes to changed every 4 or 5 years or so
If you need to vary the drum speed, mount the toothed pulley on a shaft between 2 pillow block sealed bearing assemblies and drive that shaft from the B & S motor, by changing pulley diameters to and from the engine to the toothed pulley shaft.. you can then settle on a medium size pulleys that allow you to vary the engine RPM and thus vary the drum speed depending on the type of material you are working... an important part sometimes over looked with DIY projects...as clay type material needs a much slower speed and sandy requires a much faster speed...and always a minimum of 1 to 2 hour power gas engine larger than what is minimally required so you can still get good drive power at an idle speed, which will give you a good wide span of RPMs to be able to vary the trommel drum...from idle to max RPM speed...
Hope that helps...
Klondike...