So I did some reading on this subject too. I haven't found a clear answer yet but, have some interesting facts.
First, the problem with a lot of the advice you'll get about bridge material is that it is often based on apples-to-oranges comparison. Because a bridge, like body wood, is a passive component, bridges can only attenuate a signal by dissipating vibratory energy as heat. So if it were true that brass bridges gave you "bigger sound" or more sustain, it would be because the aluminum version of an identical bridge was losing more energy than the brass one.
Comparing bridges of different design is a problem because bridges can dissipate energy in other ways too. The interface of the saddle and the bridge base plate is usually not rigid. I know that many bridges have locking saddles, which would serve to reduce vibration between the saddle and base plate. In a non-locking bridge, the saddle is not directly coupled to the base plate, and not being rigid, can spend some energy. I would expect the mass of the saddle to be more important than in a locking bridge.
Also the interface between the bridge base plate and the body is usually a screw joint- these would deform under vibration just like a neck/body joint in a bolt on bass. Ideal bridges are completely coupled to the body, so the mass of the endpoint that the string "sees" is the combined body/bridge mass- You could have a lighter bridge and a slightly more massive body, or vice versa, and it should work out pretty similarly. For differently designed bridges, however, the internal friction of the base/body interface could vary wildly- the number of screws, placement of the screws, etc. If the joint is inefficient, sustain would be reduced accordingly.
The other way a bridge could spend energy would be through internal friction. Aluminum has a specific stiffness (the ratio of stiffness to mass) around twice that of brass. Their internal frictions are about the same, so if the bridges were joined to the body in the same way (i.e. they lost no energy through their saddle/base or base/body interfaces at the same rate) I would expect them to dissipate energy as heat at about the same rate. So although brass is more massive, aluminum makes up for it by deforming less under vibration.
The bridge can also act to couple strings, so that energy in one string is spent creating sympathetic vibration in neighboring strings. I don't know what materials factors contribute to this. As JP pointed out, new style bridges that use entirely separate tailpieces for each string eliminate this coupling and the plucked string will lose less of its energy when neighboring strings begin to resonate.
So anyhow, I think this is another case where you have to look at design first, then material. Going from a flimsy factory bridge to something huge like a Badass is almost certain to sound at least a little different, the saddles and base plate are arranged differently and are considerably different in design. However it would be a hasty conclusion to decide that it was the bridge material making the difference, when there are many more fundamental differences in the design. However if you're sticking Hipshot A or B bridges and just varying the material, there isn't anything special about the brass that would cause it to behave much differently.
My guess is the reason that aftermarket bridges were cast out of brass to begin with was that it was easier at the time to cast brass. In general, they are of all-around higher quality construction than the factory jobbies. Take this all with a grain of salt, mind you. I'm not a metallurgist, just curious.
To do this right, it would be necessary to measure some very carefully generated samples through an FFT analysis program like Spectra. Human ears are very, very bad at picking out subtle differences, because they are attached to a brain that will judge sounds differently when it knows that the materials have changed
I guess I'm just to lazy or too bad at english to make such long written analysis !!
