You definately want to avoid having anything leak/drip from the truck onto the road if you're hauling it any distance. Typically you'd stack the sediment and allow it to drain and dry somewhat before hauling it off.
If you can leave on site that's definately a plus - you don't mention how much sediment you'd be digging out but keep in mind you'll want to be 100% sure that if you leave on site that it doesn't run back into the ditch or retention pond, you'll have to put up silt fence, seed and straw, etc.
Typically you can only haul off about 1/2 the capacity of a truck (ie 20 yrd bed can haul about 10 of wet/runny silt) so keep that in mind as well.
I'd say if you are going to dig out under 50 yards I'd haul it off - any more than that I'd look at leaving it onsite (if possible).
Just my 2 cents which is worth about a penny.