To amplify what is said by sstorkel, above, part of the fitting recommendations for saddle fore-aft position include the "hands up" test. If your saddle is positioned well, you should be able to lift your hands up off the handlebar (or up off the brake hoods) without having to strain too much. This wouldn't be possible if a significant portion of your body weight was being supported by your arms/hands.
I don't know what percentage of body weight IS being typically supported by the arms/hands, but would doubt if it is even as much as 10%. (Actually, it would be interesting to see some real data on this.)
The largest portion is being supported by the saddle, I think, with a significant amount being supported by the legs/feet on the pedals. This is just a guess, but I'm thinking something like 60% saddle, 30% pedals, 10% handlebars.
In terms of weight distribution between the two wheels, I think a 60/40 split is not atypical, although I've seen some people report that they've actually tried to measure this themselves, using home scales, and that they have seen more like 65/35 (rear/front).
But the weight on the saddle is not all on the rear wheels - some portion of saddle weight is transferred to the front wheel. Even more so for the weight on the pedals. And then the weight on the bars will be mostly on the front wheel, with some very tiny portion going to the rear. With enough measurements you could figure out the force vectors between the saddle, pedals, and handlebars as inputs, and the rear and front wheel as outputs, and compute the rear/front contribution for each input, and the relative contribution of each input. (You'd also need to figure out how the bike's own weight transferred between the rear and front wheels.) But then you'd be an "enginerd", like me.... :-)