Silver vs. Stainless Steel
So, I have heard from several places that Silver shakers are superior to Stainless Steel shakers because they cool drinks down faster. Looking up some number, Silver has a significantly higher Thermal Conductivity (429 vs 20ish W/M*K). Silver also has a specific heat about half (233 vs 500 J/Kg*K). I believe that there is a direct correlation with thermal conductivity and the heat transfer coefficient.
One argument I heard was that someone touched both the silver and the steel shakers, and the silver one felt colder. That is obvious due to the much higher thermal conductivity.
So, the point of all this, is that there are two situations, an equilibrium one, and a non-equilibrium one.
Equilibrium situation is where in the end, everything has the same temperature of 0 degrees Celsius. This will be reached much faster in the Silver case due to the higher thermal conductivity. Less heat will enter the drinks through the silver, and the less of the ice will have melted. In this case, it is obvious that Silver keeps the drink "colder" in that there is more solid ice.
Non-equilibrium would be say the stuff was allowed to sit for a certain amount of time. In that time, heat would have left the silver at a much much faster rate. In that same time, less heat has left the steel due to the much lower thermal conductivity. In this case, the Steel has left the drink cooler because the silver has put more energy into the liquid. In fact, finding something which is a great insulator, such as glass, might work even better because it transfers heat into the liquid at an even lower rate.
So, we come off with two conclusions. If we assume that most bartenders shake to the point where everything is all freezing cold, definitely true for things of exceptionally long shaking times (Ramos Gin Fizz), then obviously silver is better. What if they do only a few shakes, perhaps glass would be better?
I gotta throw a wrench into the works... since you're working with alcohol, equilibrium doesn't happen at 0° C. I did some of my own research when shaking Mai Tais and found that 10 seconds of good shaking brought the temperature of the liquid down to about -7° C.
I think the places that told you silver shakers are superior to steel ones were trying to sell you on a ridiculously expensive worthless upgrade to your home bar. The metal doesn't cool down the drink, the ice does! Since the silver has higher thermal conductivity, it will feel colder than the steel, yes. However, since you're shaking everything for only about ten seconds, I don't think it's going to matter one bit whether your shaker is metal, glass, or plastic. At the bar where I work, I use a wide Boston style shaker made out of stainless steel, and can use both a double rocks glass or a normal width Boston shaker metal part for the second piece. Whether I use the glass or the metal, it takes about the same amount of time for the drink to get down to the right temperature. The difference between steel and glass is much greater than between steel and silver, so I'm going to say the difference in time to chill down a drink between the two metals will be negligible at best. Taking the high price of a silver shaker into consideration (and my opinion that while the cobbler style shaker looks pretty it doesn't hold a candle to a Boston shaker), I'll stick with the $5 Boston shaker and glass set I got at a restaurant supply store, and use the money I would have spent on the silver shaker to further embellish my home bar.
You don't want thermal conductivity. This means heat transfer. You want non-thermal conductivity so that the heat stays out and the cold stays in to chill the drink faster. The best shaker to chill a drink fastest is one than is the best insulator, not thermal conductor.
I have had both over the years and frankly the cold thing is negligible. silver looks nice but will have to be polished eventually. steel is MUCH less likely to ding if (when?) dropped...
a more important consideration is to have dry hands so the condensate doesn't freeze the skin to the metal.